An Update on the Computational Theory of Hamiltonian Period Functions

University of Arkansas, Fayetteville ScholarWorks@UARK

Theses and Dissertations

12-2020

Bradley Joseph Klee University of Arkansas, Fayetteville

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Citation Klee, B. J. (2020). An Update on the Computational Theory of Hamiltonian Period Functions. Theses and Dissertations Retrieved from https://scholarworks.uark.edu/etd/3906

This Dissertation is brought to you for free and open access by ScholarWorks@UARK. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ScholarWorks@UARK. For more information, please contact [email protected]. An Update on the Computational Theory of Hamiltonian Period Functions

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Physics

Bradley Joseph Klee University of Kansas Bachelor of Science in Physics, 2010

December 2020 University of Arkansas

This dissertation is approved for recommendation to the Graduate Council.

William G. Harter, Ph.D. Committee Chair

Daniel J. Kenneﬁck, Ph.D. Salvador Barraza-Lopez, Ph.D. Committee Member Committee Member

Edmund O. Harriss, Ph.D. Committee Member Abstract

Lately, state-of-the-art calculation in both physics and mathematics has expanded to include the field of symbolic computing. The technical content of this dissertation centers on a few Creative Telescoping algorithms of our own design (Mathematica implementations are given as a supplement). These algorithms automate analysis of integral period functions at a level of difficulty and detail far beyond what is possible using only pencil and paper (unless, perhaps, you happen to have savant-level mental acuity). We can then optimize analysis in classical physics by using the algorithms to calculate Hamiltonian period functions as solutions to ordinary differential equations. The simple pendulum is given as an example where our ingenuity contributes positively to developing the exact solution, and to non-linear data analysis. In semiclassical quantum mechanics, period functions are integrated to obtain action functions, which in turn contribute to an optimized procedure for estimating energy levels and their splittings. Special attention is paid to a comparison of the effectiveness of quartic and sextic double wells, and an insightful new analysis is given for the semiclassical asymmetric top. Finally we conclude with a minor revision of Harter’s original analysis of semi-rigid rotors with Octahedral and Icosahedral symmetry. Acknowledgements

This dissertation is the refuge of a person who continues to survive cyclic nature, physical and mental, reason or no, with a True purpose kept intact. An author’s role is to transfer ideas through writing, and nothing is ever written in a perfect vacuum. In this present work, many supportive people deserve to be thanked again for their contributions. Originally I benefited from good parents, Kevin and Shiela Klee, who chose to emphasize classroom education and exploratory naturalism. My siblings, Chris and Alyssa, what fun is any outing without you two? Now we are even a bigger family with the addition of Greg Graves. Congratulations Greg and Alyssa. My feelings in the pines, at least we have this memory, all six of us traversing the mountains together, awesome! To my aunts, uncles, cousins, and grandparents, too many to name—love you all, thanks again! Good fortune also gave to me a long succession of appreciable teachers—At Olathe East Highscool, mathematician Lesley Beck, veteran Socratist Tom Fevurly, and musician David Sinha—At University of Kansas, honorable perennials, including Stan Lombardo and (more recently) Judy Roitman, Ann Cudd, Chris and Marsha Haufler, and many more; scientists, Phil Baringer, Adrian Melott, Sergei Shandarin, Hume Feldman, Craig Huneke, Dave Besson, K.C. Kong, and John Ralston—Online, Ed Pegg, George Beck, Stephen Wolfram, Jörg Arndt, Neil Sloane, Rich Schroeppel, and William Gosper—At University of Arkansas, many of the physics and math faculty, most importantly committee members Daniel Kennefick, Salvador Barazza-Lopez, Edmund Harriss, and Chair William Harter. When family members and school teachers could not console or contain me, I also had good fortune to meet a long and ever-changing cast of friends, again too many to name. You should know who you are and that you have my sincere gratitude for the important roles you played. Didn’t we try to have a community? We may have failed, but we learned something important along the way. Even then, there were other friends and allies, beyond our range of meetings—historical persons, movie stars, musicians, artists, poets, monks, nuns, etc. They also deserve our thanks and praise for how they have contributed at long distance. Inspiration is the condition for more inspiration. Without continued inspiration, no more appreciable work can ever happen. Will expiration become extinction all too soon? That is the game we now have to play, even as part-time opponents if the whole world goes wrong (and I’m worried that it has). Incidentally, thanks again to Erica Westerman, Ashley Dowling, and Ken Korth at the Arkansas Arthropod Museum. The insect trap show ”SCARABAEIDAE SINISTRAL” went on through Summer 2020, with good reviews around town. Thanks also to Fayetteville Natural Heritage Association, Pete Heinzelmann and Barbara Taylor, nice work conserving the local parks! Many beetles are still alive and well at Mt. Kessler and elsewhere, and we hope their families live long lives. Despite the 2020 outbreak of Murder Karma, at least we can hope that our scientific collection of chiral-reflecting beetles will ultimately contribute to further interest in conservation efforts. A big, completely impressed, great show, and thank you to all the following entertainers (now deceased): Cotinis Nitida, Pelidnota punctata, Osmoderma subplanata, Euophoria sepulcralis, Trichiotinus piger, even the hated Popillia japonica, genus onthophagus, Phaneus vindex, etc. Even in death, you are teaching us more about circular polarized light! We also like to thank the entire gamut of Arkansas Papilionidae for their beautiful colors, and the Cicadas for their calming rhythms, and don’t forget the fireflies. At Lake Sequoya, thanks to the wandering cows for leaving the dung pats, and to the mud, for making us a Nelumbo nembutsu to say. Thanks again citizen scientists on iNaturalist.com for photography and GPS coordinates, great work. Finally, The University of Arkansas was especially generous with time and resources so that this work could be completed. Over so many years here I was supported by a Doctoral Academy Fellowship, and even more payment was given by the Physics department as necessary and when appropriate. For administration, thanks to Reeta Vyas, Surendra Singh, Julio Gea-Banacloche, Pat Koski, Melissa Harwood-Rom, Vicky Lynn Hartwell, and others. Now that we have released the final product, I guess we can debate how well its value compares to its cost in dollars and suffering. Contents

1. Prelude to a Well-Integrable Function Theory 1 1.1. History and Introduction ...... 1 1.2. Ellipse Area Integrals ...... 7 1.3. The Kepler Problem ...... 11 1.4. Ellipse Circumference ...... 15 1.5. Elliptic Curves ...... 21 1.6. Example Calculations ...... 24 1.7. Comparing Certiﬁcates ...... 28 1.8. Prospectus ...... 31

2. An Alternative Theory of Simple Pendulum Libration 40 2.1. History and Introduction ...... 40 2.2. Preliminary Analysis ...... 46 2.3. Phase Plane Geometry ...... 50 2.4. Incorporating Complex Time ...... 58 2.5. Experiment and Data Analysis ...... 68 2.6. Conclusion ...... 74

3. Geometric Interpretation of a few Hypergeometric Series 76 3.1. History and Introduction ...... 76 3.2. Creative Combinatorics ...... 83 3.3. Diagnostic Algorithms ...... 92 3.4. Finishing the Proof ...... 99 3.5. Periods and Solutions ...... 107 3.6. A Few Binomial Series for π ...... 112 3.7. Conclusion ...... 113

4. Developing the Vibrational-Rotational Analogy (Using methods from Creative Telescoping) 117 4.1. History and Introduction ...... 117 4.2. Rigid Body Rotation ...... 121 4.3. Looking To Quantum ...... 129 4.4. The Semiclassical Leap ...... 138 4.5. Rigid Rotors Redux ...... 147 4.6. Quantum Symmetric Tops ...... 155 4.7. Conclusion ...... 162

5. Coda (on Prajna) 165

Bibliography 174

A. Supplemental Materials 181 1. Prelude to a Well-Integrable Function Theory

Quite often in physics we encounter a question about nature, which needs to be answered by taking an integral. A formalism for writing such integrals does not guarantee quality answers nor appreciable progress. Difficulties abound, especially when working with function-valued integrals, whose integrands involve one or more auxiliary parameters. Yet such parameters allow differentiation under the integral sign, so can be turned into an advantage. In many cases, a difficult-looking integral function is also the solution to a relatively simple ordinary differential equation. Playing through a few fundamental problems about ellipses and elliptic curves, we begin to hear intertwined themes from physics and mathematics. These themes will recur in more substantial followup works.

1.1. History and Introduction

Lest we look all the way back to the geometric works of antiquity (circa 200-300BC), it seems unlikely that we could find a better starting place than the musical works of Johannes Kepler (1571-1630). Kepler advanced the heliocentric theory by refining it to maximum-available precision. He did not do so by over-specializing in data analysis, rather by accomplishing superlative mastery of the quadrivium—a generalist curriculum of medieval Europe, one that placed arithmetic, geometry, astronomy, and music on even footing. As continental Europe transitioned into the brutal thirty-years war (1618-1648), Kepler published his brilliant assay in two parts, first in Astronomia Nova (1609) and subsequently in Harmonices Mundi (1619). Despite hundreds of years elapsed, Kepler’s three laws are still remembered today1:

I. The orbit of a planet is an ellipse with the Sun at one of the two foci.

II. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

1These are quoted verbatim from Wikipedia, see: ”Kepler’s Laws of planetary motion”. 1 III. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Kepler’s original ”proof” of the three laws relied upon a beautiful but doubtful musical analogy. More development was both desirable and necessary, so Kepler’s laws gave way to the Kepler Problem. It asks for a derivation of the three laws from a more fundamental physical theory, and subsequently for an adherent solution of the time-variant planetary equations of motion. This task can not be accomplished by harmony alone. It requires another paradigm change, which first came about during the European enlightenment2. With the publication of Philosophiæ Naturalis Principia Mathematica (1687), Isaac Newton (1642-1727) made a significant contribution toward the initiation of European Enlightenment. In this effort to defeat the specter of irrational religiosity, Newton’s work was like a clarion, calling all subsequent generations to the front lines of scientific research. Newton’s three laws are also remembered to this day:

I. Absent of an external force, an object in motion stays in motion, while an object at rest stays at rest.

II. A net force F applied to a massive object causes an acceleration a. The two dynamical variables are linearly proportional by the mass m, i.e. F = ma.

III. For every force from one body to another, there is an equal and opposite response force

from the later body to the former (It is often written, F21 = F12). −

To these three, Newton also gave an important addendum regarding the particular case of gravitating bodies, the Universal Law of Gravitation,

G. The attractive force between two point masses is directly proportional to the product of masses, and is inversely proportional to the square of the distance between them.

2Do not confuse European and Asian enlightenment! These movements happened at diﬀerent times, in diﬀerent geographical regions. (When considering whole and indivisible spacetime, confuse it all!) 2 If m1 and m2 are the masses, and r the distance vector, the gravitational force vector F

m1m2 is usually written F = G r·r rˆ, with gravitational constant G. The adjective ”universal” indicates that law G applies to the orbits of planets, to the orbit of the moon, to the tides between the moon and the oceans, as well to the oscillation of various types of mechanical pendulums. In fact, universal law G applies to any pair of gravitating bodies, anywhere in the universe3. Accepting I, II, III, and G as all valid and applicable, Kepler’s laws can be proven mathematically using only the geometrical techniques of Newton’s day and age. Richard Feynman (1918-1988) took this as a challenge when he gave a lecture on planetary motion, March 13, 1964 [34]. The lecture stands on its own as an active and imaginative contribution to the history of science, and it is quite different from anything that we would readily recognize as a typical solution to the Kepler problem. Famously, Newton wrote ”if I have seen further it is by standing on the shoulders of Giants.” In so doing he became a part of the gigantic scientific enterprise, as did his follower Leonhard Euler (1707-1783). Perhaps no one worked more than Euler to raise this giant into its present-day stance. Encyclopedic works typically credit Euler for originating (or at least co-originating) the first abstract definition of what a function is, and for giving the first important examples4. Most noteably, the functions ex, cos(x), and sin(x), were written by Euler in series expansion,

1 1 1 1 ex = 1 + x + x2 + x3 + x4 + x5 + etc. 2 2 3 2 3 4 2 3 4 5 1 1 · · 1· · · · cos(x) = 1 x2 + x4 x6 + etc. − 2 2 3 4 − 2 3 4 5 6 1 · · 1 · · · · 1 sin(x) = x x3 + x5 x7 + etc. − 2 3 2 3 4 5 − 2 3 4 5 6 7 · · · · · · · · · with ”+etc.” indicating continuation of the numerical pattern to inﬁnity. From this deﬁnition

3We are not disregarding Einstein’s theory of general relativity, so must also say that Newton’s universal laws are not exactly universal. In some parts of the universe, they completely fail, e.g. black holes. 4Be careful if studying Wikipedia. The article ”History of the function concept” at least needs a section on pre-history, starting with compass and straightedge, the functional implements of antiquity. Also, don’t forget to read primary source documents. Hundreds of Euler’s works are available online through the Euler archive[28], see E101 Ch. 7-8 for early deﬁnitions of ex, cos(x), and sin(x) and more. 3 (log-linear) N IIIIII ENVELOPE CALCULATION 106 N III Mortality Rate: R = 1/100 106 Time to Double: τ = 4 days II 104 Infection → Death: ∆t = 20 days I 102 NIII R−1 2∆t/τ NI ≈ infected t (days) 3200 death 0 ≈ 20t0 40 t (days) 20t0 40 source: twitter: @ColinTheMathmo LEGEND: I. Deaths II. 100 × Deaths III. Infection Estimate see also: Insect Pandemics in U.S.

log(2) t Fig. 1.1.: Onset of a pandemic: exponential curves N(t) = N0e τ plotted over time t.

it is straightforward to infer all the following annihilating relations,

x x 2 2 ∂xe e = 0, ∂ cos(x) + cos(x) = 0, and ∂ sin(x) + sin(x) = 0. − x x

The important composition identity ex = cos(ix) i sin(ix), where i2 = 1, also follows, − − as does the beautiful and profitable Euler’s identity that eiπ = 1. It is apparent from his − collected works that Euler understood the practical value of transcendental functions, and intended for subsequent generations to use these tools to continue solving new and interesting problems. The three functions ex, sin(x) and cos(x) are among the best specialized tools a scientist ever receives. When used together with statistical analysis, these tools are often enough to predicate an entire career, even in practical disciplines or the so-called ”real world”. The infographic Fig. 1.1, gives one example related to the COVID-19 pandemic of 2020. Meanwhile, sine and cosine contribute an essential part to subsequent analyses. Euler was also interested in calculus as a theory, regardless of the material or the mundane. He thought abstractly, made numerical analogies, and ventured into lesser known realms of mathematics to find and analyze other important functions. The Euler archive records early series definitions for elliptic integrals,

2 1 12 3 12 32 5 E(x) = 1 x · x2 · · x3 etc., π − 22 − 22 42 − 22 42 62 − 2 12 12· 32 12· 32· 52 K(x) = 1 + x + · x2 + · · x3 + etc., π 22 22 42 22 42 62 · · · 4 under entry numbers E028 and E503 respectively5. These basic examples eventually led Euler to an early discovery of the general hypergeometric series, in his notation,

ab Y (a + 1)(b + 1) Y (a + 2)(b + 2) s = 1 + x + x2 + x3 + etc., 1 c 2 (c + 1) 2 (c + 2) · · · where recursive symbol Q stands for multiplication by the previous series coefficient. This equation appears verbatim in Specimen transformationis singularis serierum, archive entry E710, alongside its defining differential equation

”0 = x(1 x)∂∂s + [c (a + b + 1)x]∂s abs”. − − −

We no longer use Euler’s notation or ordering, and instead write an annihilating operator,

2 F = z(1 z)∂ + c (a + b + 1)z ∂z ab A − z − − 2 such that F F = z(1 z)∂ F + c (a + b + 1)z ∂zF abF = 0, A ◦ − z − − which constrains all possible solutions. The putative simplest series solution, a, b X n 2F1 z = fnz with f0 = 1 and (n + 1)(n + c)fn+1 = (n + a)(n + b)fn, c n≥0 introduces a concise notation where, for example, elliptic integrals are easy to deﬁne,

1 1 1 1 2 2 , 2 2 2 , 2 E(z) = 2F1 − z and K(z) = 2F1 z . π 1 π 1

However nice it may be to get rid of Euler’s ”etc.”, the simple hypergeometric solution is not a unique or final definition. Sections 4 and 5 of this chapter will explore alternative instantiations of E and K, targeted toward precise and efficient calculation. Reversing the order of presentation, we mean to portray the hypergeometric differential equation as more fundamental than any one particular solution6. This reversal raises a question about procedure: if 2F1 is to follow from F , what shall precede F ? For special A A

5The notation here is similar, not identical, to notation used originally by Euler. Standard usage of letters K and E is a more recent development attributed to A.M. Legendre (1752-1833). 6In fact, the second-order H.D.E. must have a solution-space with two degrees of freedom. 5 values (a, b, c) it is possible that F has a natural geometric origin[61]. This is the case for A functions E and K, which may also be written as,

Z π/2 Z π/2 1 E(z) = p1 z sin(φ)2 dφ and K(z) = dφ. p 2 0 − 0 1 z sin(φ) − Euler already knew how to derive these integral forms (or similar) from geometry and/or Newtonian physics, and he took them as a fundamental starting place. However, Euler did not have a rigorous procedure for analyzing partial derivatives of the integrands, so he could

not derive the corresponding cases of F without resorting to series expansion methods. A In present times, the related fields of Periods, Creative Telescoping, and Holonomic Functions can add rigor where it may be missing7. Algorithms from these theories help to analyze the sort of integrals typified by elliptic E and K. For simplicity sake, let us take the one-dimensional case, where I(α) = H dI dt over domain (α), an algebraic X (α) dt X plane curve, also a Jordan curve8. Integral I(α) is sensitive to how the shape of curve (α) X depends on the auxiliary parameter α. If (α) and dI/dt are both sufficiently simple, then X there will exist an annihilating operator I Q[[α, ∂α]] (also called a ”telescoper”), which A ∈ dI d t satisfies I I(α) = 0 because I = (Ξ ). An annihilator I and its certificate A ◦ A ◦ dt dt I A t ΞI can sometimes be calculated concurrently using only a combination of partial-fraction decomposition and the Ostrogradsky-Hermite reduction. This is the case for elliptic E and K, as well for many other geometries to appear in our sustained research effort.

An expression such as I I(α) = 0 tells us that I(α) is the solution of an ordinary A ◦ t differential equation. How should we understand certificates such as ΞI ? Can we calculate certificates, and should we? If so, how? These are motivating questions for the present work. Using an empirical, example-driven style, we will go from Kepler’s laws and Newton’s laws in Sections 2 and 3, to ellipses, elliptic curves, and elliptic integrals in Sections 4 and 5, while stopping only briefly to solve a few problems in Section 6. Finally in Section 7, we take a closer look at certificate geometry. As with any prelude, the progression from start to start

7For background and overview, see references [4, 21, 56, 61, 62, 91]. 8See also Mathworld: Algebraic Curve, Jordan Curve. 6 0 Ae(P1) A (P1) A(P1)

re1 r1 y1 Pa θe1 θ1 P0 x1 → 1 ←

2e ←− −→

Fig. 1.2.: An Ellipse with e = 2/3. E

is a right of passage, a test of technical skill, and ultimately only a hint of what is to come next. Rather than concluding entirely, Section 8 gives the prospectus to a dissertation where physical and mathematical themes will cipher ’round again.

1.2. Ellipse Area Integrals

Kepler’s second law asks for the area swept out by a point moving on the circumference of an ellipse, from P1 = (x1, y1) to P2 = (x2, y2). We choose all points from the ellipse,

= (x, y) : (1 e2)(x + ae)2 + y2 = a2(1 e2) , E { − − } with eccentricity e [0, 1) and semi-major axis length typically set to a = 1 (without loss ∈ of generality). These conventions for ellipse place one focus at the origin and another E at x = 2e, as in Fig. 1.2. By the integral property that R P2 dA = R P2 dA R P1 dA, a − P1 P0 − P0 standard reference point P0 can be chosen to simplify analysis. Either the apogee or perigee is a natural choice. Between the two, we choose the perigee at P0 = (x0, y0) = (1 e, 0). In − Cartesian coordinates, the trigonometric area integral,

Z P1 Z 1−e Z 1−e x x p 2 2 AE (P1) = dAE = y dx = (1 e )(1 (x + e) ) dx, P0 x1 x1 − − 7 has a simple and well-known closed-form,

x 1 2 A (P1) = √1 e arccos(e + x1) (e + x1)y1 . E 2 − −

x However, area AE (P1) is the Cartesian area, so does not immediately help with the Kepler problem. Instead we need to calculate the sectorial area,

Z P1 Z θ1 Z θ1 2 2 θ θ 1 2 1 1 e AE (P1) = dAE = r dθ = − dθ, P0 2 0 2 0 1 + e cos(θ)

θ in polar coordinates where x = r cos(θ) and y = r sin(θ). The integrand of AE (P1) is prohibitively complicated9. From Fig. 1.2, sectorial area equals to Cartesian area plus or

θ x 1 minus the area of a triangle with base length x1 and height y1 , A (P1) = A (P1) + x1y1 | | | | E E 2 (x1 is negative to the left of the origin). There is another way to derive this identity, but without using trigonometry. The following exercise in calculus may seem superﬂuous, but it is worthwhile training when the goal is to progress to more complicated integrals, as in subsequent sections of this paper.

Choosing P1 as the apogee point Pa = (xa, ya) = ( 1 e, 0), the total ellipse area is written, − − θ 2 A(e) = 2A (Pa) = π√1 e . Complete area A(e) is an algebraic function, which satisfies E − 2 2 A A(e) = (1 e )∂e + e A(e) = (1 e )∂eA(e) + eA(e) = 0. Moving the annihilating A ◦ − ◦ − 2 θ x operator A = (1 e )∂e + e under the integral sign of either A (P1) or A (P1) we obtain A − E E two checks on the validity of A, A θ θ 2 2 dAE dΞA 3 1 2 2 (1 e ) A = = e + cos(θ) + e cos(θ) − , A ◦ dθ dθ − 2 2 (1 + e cos(θ))3 x x 2 2 dAE dΞA (1 e ) (x + e) A = = − − . A ◦ dx dx p(1 e2)(1 (x + e)2) − − θ x 10 Certificate functions ΞA and ΞA must exist and contribute to an exact differential so that

H θ H x A dA and A dA will equal zero, as necessary. The certiﬁcate functions follow A ◦ E A ◦ E

9 x Mathematica automatically evaluates AE (P1) within a few seconds, but takes a long time thinking on θ AE (P1), and returns an over-complicated answer[54]. 10Exact diﬀerentials integrate to zero on any complete cycle, H df = 0 implies df exact. 8 from indeﬁnite integration11,

Z dΞθ 1 e2 2 Ξθ = A dθ = sin(θ)1 + e cos(θ) − , A dθ − 2 1 + e cos(θ) Z dΞx Ξx = A dx = (1 e2)p(1 e2)(1 (x + e)2). A dx − − −

θ x Upon another integration to P1, in terms of ∆A(e) = A (P1) A (P1), we have that E − E Z θ1 θ Z 1−e x 2 dΞA dΞA (1 e )∂e∆A(e) + e∆A(e) = dθ dx. − 0 dθ − x1 dx

θ x Terms on the right-hand side are evaluations of ΞA and ΞA,

Z θ1 θ Z 1−e x dΞA θ e dΞA x dθ = ΞA(P1) = (r1 + 2 x1)y1, and dx = ΞA(P1) = (r1 + ex1)y1, 0 dθ − x1 dx −

12 while the term ∂e∆A(e) may safely be ignored as equal to zero . Putting it all together,

θ x 1 1 ∆A(e) = Ξ (P1) Ξ (P1) /e = x1y1, we ﬁnd again the product x1y1. Taken separately A − A 2 2 the certiﬁcates seem like nothing too special. Combined via subtraction, they are a circuitous means to determine the green triangular area of Fig. 1.2. We will return to this idea in Section 7, but presently need to continue solving. Yet another important coordinate system exists, the Keplerian coordinates,

x = cos(ϑ) e and y = √1 e2 sin(ϑ), − − written in terms of the eccentric anomaly ϑ, also the polar angle of Fig. 1.3. Introducing

θ 2 the mean anomaly, Θ(P1) = 2A (P1)/√1 e , and combining various equations, we ﬁnally E − arrive at Kepler’s equation Θ = ϑ e sin(ϑ). If instead we measure sectorial area from the − second focus, Kepler’s equation would read Θe = ϑe+ e sin(ϑe). Choosing Θe = Θ, the points − P (Θ) and P (Θ)e fall onto the intersection of ellipse with a sine wave, E

11If these integrals are too difficult by the usual deductive procedures, try a guess-and-check strategy. 12 Point P1 is fixed, thus x1 and θ1 are fixed. Variation of e by de allows for disagreement on a triangular 2 area, ∆A(e + de) ∆A(e) de , thus ∂e∆A(e) = 0. See also Section 7 and Fig. 1.11. − ∝ 9 1 1 2 Θ(e P1) 2 Θ(P1)

Pa ϑ1 P0 → 1 ←

2e ←− −→

Fig. 1.3.: Keplerian Coordinates.

π e = 2/3 ∆Θ = 5

√1 e2 − π ←− e −→ Fig. 1.4.: Asymmetric Intersection Geometry for P (Θ),P (Θ)e = (Θ) . { } W ∩ E

10 ( ! ) e y (Θ) = cos + Θ e, y : y R , W √1 e2 − ∈ − traveling at constant velocity vy dy/dΘ along the y-axis, as in Fig. 1.4 (for more details, ∝ see ref. [33]). This geometric fact, though neat, is not entirely satisfying from a physicist’s point of view.

1.3. The Kepler Problem

Classical mechanics determines planetary orbits according to a gravitational force field, which sums over contributions from all masses within a particular region of space. As the masses move through space, generally the gravitational force field changes with time. However, when one body dominates the gravitational field we may assume a time-independent force field where the dominant body has zero velocity [51, 65]. In the Kepler problem, a star of mass M generates a fixed gravitational field, which determines the classical orbit of a planet whose mass m satisfies m M, as in Fig. 1.5. We choose a system of cylindrical coordinates which places the star at the origin, and then we can solve the Kepler problem by combining

Newton’s law of gravitation, F = GMm rˆ, with his second law of motion, F = d p. − r2 dt Isotropic symmetry around the sun immediately suggests a few well-known shortcuts. Throughout time, an isolated, classically orbiting planet falls into a plane (x, y, z): z = 0 { } with normal vectorz ˆ r ˙rdt. The necessary condition,z ˙ = 0, follows from conservation ∝ × d of momentum along the vertical, Fz = 0 = dt pz. The in-plane angular component

Fθ of gravitational force F also equals zero, thus conservation of angular momentum, d 2 ˙ rFθ = 0 = dt Lθ, constrains angular motion by Lθ = mr θ = L0. The vector identity

L = r p = m(r r˙) = L0zˆ conceals a hint to Kepler’s second law. Vector ˙r dt translates × × r along a tangent line, r2 = r1 + ˙r1dt. In the inﬁnitesimal limit, sectorial area takes a

1 1 1 L0 dA 1 L0 triangular shape such that dA zˆ = r1 r2 = r ˙r dt = dt zˆ, or = . 2 × 2 × 2 m dt 2 m GMm ˙2 Newton’s law along the radial dimension, Fr = 2 + mrθ = mr¨, has an extra term − r for the ﬁctitious force of centripetal acceleration. A standard approach substitutes for θ˙ and

11 1 L0 20 m T

Fig. 1.5.: A Strong Gravitational Force Field.

changes variables by r u = 1/r and dt dθ = L0 u2dt. Then we obtain a recognizable → → m d2u GMm2 form, 2 +u = 2 = c0, essentially the deﬁning diﬀerential equation for sin(θ) and cos(θ). dθ L0

The general solution is written as u = c0 + c1 cos(θ) + c2 sin(θ), with initial conditions c1 and c2. In terms of radial coordinate r the solution becomes 1 = r c0 + c1 cos(θ) + c2 sin(θ) , or

2 2 2 2 in Cartesian coordinates x = r cos(θ) and y = r sin(θ), (1 c1x c2y) = c (x + y ). This − − 0 x-y constraint equation has no terms higher than quadratic, so its locus of points determines a conic section. Of the conic sections, only the circle and the ellipse will bind the orbiting planet to the sun. A rotation of the ellipse is chosen by setting c2 = 0. Upon rearranging

2 2 2 2 2 2 2 2 terms we ﬁnally reach an almost-canonical form, c = (c1 + (c c )x) + c (c c )y . 0 0 − 1 0 0 − 1 Section 2 already goes through suﬃcient detail on how to calculate ellipse area integrals, so completing the Kepler solution only requires a bit of dimensional analysis. Comparing

2 ellipse constraints determines integral constants in terms of eccentricty, 1/c0 = 1 e and − 2 e/c1 = 1 e . In units of length and time where GM = 1 and a = 1, constant c0 entirely − −1/2 determines sectorial velocity dA = 1 L0 = 1 c = 1 (1 e2)1/2. The yearly period does not dt 2 m 2 0 2 − dt depend on eccentricity, for Y = dA A(e) = 2π. According to Kepler’s third law, yearly period does depend on semi-major axis length, Y (a) = 2πa3/2. This is exactly the result we ﬁnd by restoring scale a2 to total area, and a1/2 to sectorial velocity. With the three laws proven, all that’s left is to invert Kepler’s equation and make another plot or two.

12 e = 2/3

Fig. 1.6.: Error of ϑ1(Θ).

Whenever ϑ is an integer multiple of π the perturbing term e sin(ϑ) equals to zero and ϑ = Θ. The slope of the inverse function, dϑ/dΘ is then easy to determine, dϑ/dΘ = (1 e)−1 for − even n, (1 + e)−1 for odd n. These boundary conditions are suﬃcient data to build a decent ad hoc approximation. Identity ϑ(nπ) = Θ suggests the form ϑ(Θ) = Θ + f(Θ) sin(Θ), with either

e 2e Θ(2π Θ) f(Θ) = 1 − , 1 e − (1 + e) π2 −e 1 e or f(Θ) = + cos(Θ) , 1 e 1 + e 1 + e − chosen to ﬁt the slopes. The former approximation (with Θ evaluated modulo 2π) achieves 99% accuracy13 for any eccentricity satisfying e 0.5. Both approximations have better ≤ than 99.5% accuracy when e 0.25 and worse than 5% error when e > 0.7. In our solar ≤ system, Mercury has the highest eccentricity at e 0.2, so either ad hoc approximation will ≈ work just ﬁne. Calculation of planetary orbits to much greater accuracy would require the entire physical theory to be reworked with fewer simplifying assumptions. Kepler’s laws do not hold out in general, and even Newton’s laws, famously, have trouble with Mercury. In ranges of eccentricity where ad hoc approximations begin to fail, a higher precision

13 100 Here accuracy is deﬁned as ϑApproximate ϑExact , in percentages of circular circumference. 2π | − | 13 t

Y −→

x y 1 2 9 e = 0, 2 , 3 , 10 .

Fig. 1.7.: Kepler Orbits in Spacetime.

solution to Kepler’s equation is desirable and necessary. Assuming Θ a ﬁxed constant, Newton’s iterative method will ﬁnd ϑ(Θ) as root of Θ (ϑ e sin(ϑ)) = 0, according to the − − recursive equation,

Θ (ϑi e sin(ϑi)) ϑi+1 = ϑi + − − , ϑ0 = Θ. 1 e cos(ϑi) −

A ﬁrst approximation ϑ1 satisﬁes the boundary conditions above, and reaches about the same accuracy as either ad hoc approximation. Iteration to higher values of i converges the

estimate ϑi(Θ) toward it’s actual value, such that Θ (ϑi e sin(ϑi)) approaches zero, but − − only at the cost of increased complexity for the functions ϑi(Θ). Other references such as

[68] discuss convergence and error analysis, here we are content simply to use function ϑ4(Θ) and plot a few spacetime trajectories in Fig. 1.7.

14 SPECIMEN CONSTRVCTIONE AEQVATIONVM DIFFERENTIALIVM SINE INDETERMINATARVM SEPARATIONE

Fig. 1.8.: Transcription of the Latin title to Euler’s E28.

The spacetime diagram is a graphical solution of the Kepler problem. It shows isoperiodic orbits of varying eccentricity, all with ﬁxed length scale a = 1. Solutions with a = 1 simply 6 scale the vertical axis by a factor a3/2, so have entirely similar shapes. When e = 0, the worldline is a circular helix. For any other e (0, 1), the worldline is an almost-helix with ∈ anisotropic vertical stretching via ϑ(Θ). Green or blue coloring indicates where an ad hoc solution of ϑ(Θ) will or will not work decently well. A dividing wordline of e = 1/2 appears in red (also compare green e = 2/3 worldline with errors of Fig. 1.6).

1.4. Ellipse Circumference

Euler’s diﬃcult but comprehensive approach to mathematics and physics is useful even when asking seemingly simple questions, for example: What is the average velocityv ¯ of a Kepler orbit? In the special case of a circular orbit, instantaneous velocity v is a constant of motion,

−1/2 −1/2 for L0 = m a v, and this impliesv ¯ = v = a (again in units where GM = 1). The a scaling of average velocity also follows from its deﬁnition as distance-over-time,v ¯ = C0/Y ,

2 with circular circumference C0 = 2πa. The circumference C(e ) of a general ellipse, of course, depends intricately on eccentricity e, so too must average velocity,v ¯(e) = C(e2)/Y . Thus to answer the deceptively simple question about average velocity, we must follow Euler and ﬁnd the function C(e2) by arclength integration. It is not too easy a task, and perhaps not worthwhile if the Kepler problem is our only motivation. We entreat the wary reader to keep in mind the richness of nature, and to have faith that mathematics will continue to prevail in other interesting circumstances.

15 We choose coordinates14 and redeﬁne that = (p, q): p2 + (1 α)q2 = 1 α with E { − − } α = e2 and a parametric solution q = sin(ϕ) and p = √1 α cos(ϕ). The arclength integral, − already assuming a = 1, takes a concise form in terms of angle ϕ,

s 2 2 I I p I dp dq I C(α) = dl = dp2 + dq2 = + dϕ = p1 α sin(ϕ)2dϕ. dϕ dϕ − Over a complete domain, ϕ [0, 2π], term-by-term integration of the α-series expansion ∈ yields a solution,

n 2 n X 1 2nα I X 2π 2n α C(α) = sin(ϕ)2ndϕ = . 1 2n n 4 1 2n n 16 n≥0 − n≥0 − This is not the only solution of C(α), nor even the best. Practically speaking, values of C(α) become difficult to calculate at large α where convergence of the series expansion slows to a crawl. Fortunately, there is a stronger analysis, one that owes back to Euler himself. Euler was among the first to realize that the function C(α) could be defined as the solution of an ordinary differential equation. Though it produces the correct answer, his intuitive method of solution leaves some doubt and room for improvement. A more rigorous approach starts by observing that the first two α-derivatives of the arclength element dl can be written in terms of the trigonometric polynomial Φ = dl 2 = 1 α sin(ϕ)2, dϕ − 1 1 − 1 2 1 1 − 1 − 3 ∂αdl = Φ 2 Φ 2 dϕ, ∂ dl = Φ 2 2Φ 2 + Φ 2 dϕ, 2α − α −4α2 − after decomposing to partial fractions15. Each term is of the form w Φn/2, with odd n and w a ratio of polynomials in variable α. For every such integrand, the technique of Hermite reduction produces a canonical least form by the addition of exact ϕ-differentials. With u, v, and w all undetermined functions of angle ϕ, a first reduction [w] of w is written as,

[w] dv/dϕ 1 w d v = u = . Φm−1 − m 1 Φm−1 Φm − dϕ (m 1)Φm−1 − − This equation can be iterated to ﬁnd successive reductions of w, but only when u,v, and w satisfy a closure requirement. The closure requirement follows from analysis of the

14Compare with Keplerian coordinates by ϕ = π/2 ϑ and q = x + e. n − n 15By inspection, ∂ndl = ( −1 )n(2n 3)!! P ( 1)m Φ1/2−m, see also OEIS [89]: A330797. α − 2α − m=0 − m 16 consequential identity w = Φu dΦ v. Notice that Φ = 1 α sin(ϕ)2 is a quadratic polynomial − dϕ − of sin(ϕ) and an even function, while dΦ = 2α sin(ϕ) cos(ϕ) is quadratic and odd16. When dϕ − u and w are even, v must be odd. Imposing a degree bound d, we have that,

d+1 d d X X dΦ X w sin(ϕ)2n = Φ u sin(ϕ)2n v cos(ϕ) sin(ϕ)2n−1, n n dϕ n n=0 n=0 − n=1 with d + 2 coeﬃcients to powers of sin(ϕ)2 and 2d + 1 undetermined coeﬃcients on the right

hand side. Choosing d = 1, the system of linear equations is exactly solvable for u0, u1 and v1 in terms of w0 and w1, and w3 can be ignored by always requiring w3 = 0. The existence of a degree bounded Hermite reduction guarantees an annihilating operator

E for C(α) with no more than three terms. We will use a matrix method to calculate this A operator directly from the solution,

w0α + w1 2 w0α + w1 u = w0 + sin(ϕ) , v = sin(ϕ) cos(ϕ). 1 α − 2(1 α) − − Functions u and v determine a set of invariants for Φ, which collect in reduction matrices,     1 0 −α −1   0  2(1−α) 2(1−α)  U =   , V =   .  α 1   α 1  1−α 1−α 1−α 1−α

These two matrices allow us to simplify the reductive process to mere matrix multiplication,

1 0 T [w] = (U V ) w = R(m) w, with column vector w = [w0, w1] . In terms of − m−1 · · T 2 w(dl) = [1, 0] , derivatives ∂αdl and ∂αdl reduce according to, T 1 1 1 3 [w(∂αdl)] = I R w(dl) = , − , 2α − 2 · 2(1 α) 2(1 α) − − T 2 1 1 1 3 3 3 [[w(∂ dl)]] = 2 I 2 R + R R w(dl) = − , , α − 4α − 2 2 · 2 · 4(1 α)α 2(1 α)α − − with 2 2 identity matrix I. Three column vectors with two components each must admit × at least one zero-sum. In this case, the identity,

T 2 [0, 0] = w(dl) + 4(1 α)[w(∂αdl)] + 4(1 α)α [[w(∂ dl)]]. − − α 16Even functions satisfy f(ϕ) = f( ϕ); odd functions satisfy f(ϕ) = f( ϕ). − − − 17 ϕ 2 dl dΞE reveals an annihilator E = 1 + 4(1 α)∂α + 4(1 α)α∂ such that E = and A − − α A ◦ dϕ dϕ ϕ consequentially E E(α) = 0. Certificate function Ξ need not be calculated; however, A ◦ E when known, it provides a worthwhile quality check on E. Indefinite integration, A Z dl Z 1 2 sin(ϕ)2 + α sin(ϕ)4 cos(ϕ) sin(ϕ) Ξϕ = dϕ = dϕ = , E E − 3/2 p A ◦ dϕ 1 α sin(ϕ)2 1 α sin(ϕ)2 − − after careful bookkeeping, must agree with a total of exact differentials. The row vector,

cos(ϕ) sin(ϕ) α 1 V(m) = − , , 2(m 1)Φm−1 1 α 1 α − − − of function v determines the certiﬁcate by a recursive calculation, ! 2(1 α) 1 α Ξϕ = − − V( 1 ) + − 2V( 1 ) V( 3 ) V( 1 ) R( 3 ) w(dl), E α 2 α 2 − 2 − 2 · 2 ·

ϕ dl dΞE which follows from the reductions above. The zero sum, E = 0, is easy to A ◦ dϕ − dϕ ϕ check, and veriﬁes E against Ξ . Although the preceding derivation looks formidable, it is A E actually an easy, n = 2 case of a general n-dimensional method. Such calculations are not usually carried out by hand. In practice, a computer algebra system such as Mathematica routinely automates the details (Cf. Appendix A). If there is any doubt about the veracity of an algorithmic derivation, the annihilating relation can be checked again on the output.

After centuries of development, analysis and solution of E now follows a widely-known, A standard schedule: ”The regular singular points of E are correctly aligned, so that it is A possible to read out hypergeometric parameters (a, b, c) = ( 1/2, 1/2, 1), which deﬁne a − general solution around α = 0”. That solution17,

1 1 1 1 Z 1 1 −2 2 , 2 2 , 2 −1 2 , 2 C(α) = (C0) 2F1 − α + (C1) 2F1 − α α 2F1 − α dα, 1 1 1

agrees with the earlier term-by-term expansion when C(0) = C0 = 2π and C1 = 0. The

h − 1 , 1 i P n function 2F1 2 2 α = fnα sums over n 0, with coeﬃcients fn deﬁned according to 1 ≥ a hypergeometric recursion, 2 n 2 1 1 1 2n 1 f0 = 1, (n + 1) fn+1 = (n )(n + )fn fn = . − 2 2 ⇐⇒ 1 2n n 16 − 17Mathworld: Hypergeometric Function, Second-Order ODE Second Solution. 18 Table 1.1.: Constraints on ) α ( C P n E ) α ( C = cn α , n 0 A ◦ ≥ n 0 = cn = 0 0

1 32 2C 12 1C C 0 − − 2 512 C 3 320 2C + 7 0C . − . . .

n determines C n+1

Yet nothing much is gained by changing notation. The solution, so far, has not diversified enough to avoid convergence difficulty around the regular singular point at α = 1. We will forge a way forward by taking advantage of flexibility inherent to the operator E. A Change of variables α α = 1 α produces another, reversed annihilating operator, → − 2 E E = 1 4 α ∂ α + 4(1 α ) ∂α , with a differing solution ) α ( C around α = 0. A → A − − α Operator E is again hypergeometric, but it is not as easy to solve. The parameters A (a, b, c) = ( 1/2, 1/2, 0) set c equal to zero, and consequently a1 = a0/0, utter nonsense. − We resort to a second solution, similar to the first above,

1 3 1 3 Z 1 3 −2 2 , 2 2 , 2 1 2 , 2 ) α ( C = 1C α 2F1 α + 0C α 2F1 α − α 2F1 α d .α 2 2 1 α 2 − In this case, α = 0 corresponds to a completely collapsed ellipse with (0) C = 4, thus the second term can not be ignored. Rather than go into detail repeating a proof from Mathworld, let us derive the same solution using Frobenius’s method. An Ansatz that,

1 3 3 0C 2 , 2 X n ) α ( C = 0C + 1C + 0C α log( )α α 2F1 α + C n α , 8 4 2 − n>1 allows two degrees of freedom by 0C and C 1, while the other C n coefficients with n > 1 are entirely constrained by the differential equation E ) α ( C = 0, as in Table 1.1. Choosing A ◦ that 0C = 4 and 1C = 4 log(2) 5/2 defines an α-reversed circumference function such that − ) α ( C = C(α) over the domain α = 1 α [0, 1]. − ∈ 19 The appearance of log(2) in 1C is an unresolved mystery of this presentation. In practice,

1 1 the zero sum C( ) ( ) C = 0 determines 1C to an arbitrary precision, which depends on 2 − 2 N, the number of summed terms. Choosing a large value such as N = 100, we calculate

that 1C 0.27258872223978123766892848583271, with error creeping in only on the very ≈ last digit. Such precision is overkill for many use cases. Instead, the choice of N should be tuned to speciﬁc precision goals. Any value of the piecewise function,   C(α) α 1/2 C(α) = ≤ , pw  (1 α) C α > 1/2 −

1 1 is expected to reach roughly the same precision as C( 2 ) = ( 2 ) C . An N = 60 approximation −16 already reaches double precision of E C(α) < 10 on the domain α (0, 1). This check A ◦ pw ∈ assures the quality of C(α), which we can now begin to use in calculations about average pw orbital velocity or whatever else. More importantly, the process of finding an answer has introduced concepts and techniques that we will have occasion to use again, when building computable realizations for other similar integral functions. Thus far we have deliberately avoided standard nomenclature by neglecting to mention the tautology that C(α) = 4E(α), in terms of E(α), the complete elliptic integral of the second kind. In so doing, we might have skipped over another integral function, K(α), the complete elliptic integral of the first kind. There is no deductive reason why one should precede the other, for it is possible to define that either K(α) = (1 2α∂α) E(α) or − ◦ E(α) = (1 α) + 2(1 α)α∂α K(α). The reason for nomenclature to ignore historical − − ◦ ordering is apparently more subtle. In the modern theory of elliptic curves and elliptic functions, as well in the theory of pendulum motion, function K(α) is a period not too dissimilar from Kepler’s orbital period Y (a). Neither are these periods too similar. Again K(α) is hypergeometric whereas Y (a) is only algebraic. Before we get a chance to classify in more detail, we will show how elliptic integral K(α) measures a family of elliptic curves.

20 1.5. Elliptic Curves

Another worthwhile geometric problem asks for the total area S(α) within a deformable, closed elliptic curve (α). An answer to this problem contributes a key fact to the C construction of elliptic functions, or sometimes, even to an exact solution of the simple pendulum’s motion [58]. Yet various acceptable choices of (α) are not exactly equivalent C from a metrical perspective. In particular, enclosed area S(α) depends explicitly on the shape of curve (α). Pursuant to ﬁnding the integral function K(α), we will use a variant C of Edwards’s normal form (Cf. [27]) and select square-symmetric curves, n o (α) = (p, q): α = p2 + q2 p2q2 , C − with α [0, 1). A few of these curves are depicted in Fig. 1.9. The selection constraint, ∈ α = p2 + q2 p2q2, must be solved to obtain integrands of the area integrals, − √ √ s Z P0 Z α Z α 2 p p α p either S (P1) = dS = q dp = − dp C C 1 p2 P1 p1 p1 − s Z P1 Z q1 Z q1 2 q q α q or S (P1) = dS = p dq = − dq C C 1 q2 P0 0 0 − Z P1 Z φ1 Z φ1 p 2 φ φ 1 1 α sin(2φ) or SC (P1) = dSC = λ dφ = − − 2 dφ, P0 0 0 sin(2φ) with boundaries P0 = (p0, q0) = (√α, 0) and P1 = (p1, q1) = √2λ1 cos(φ1), √2λ1 sin(φ1) . The third alternative is written in action-angle coordinates18 (λ, φ), deﬁned relative to Cartesian (p, q) by p = √2λ cos(φ), q = √2λ sin(φ), or relative to polar coordinates (r, φ), by

1 2 λ = 2 r , φ identical. After perigee P0, the next nearest apogee Pa falls onto a diagonal line p of symmetry where pa = qa = 1 √1 α or φa = π/4. According to dihedral symmetry, − − p q φ this choice determines the total area S(α) = 8SC(Pa) = 8SC(Pa) = 8SC (Pa). Diﬀerentiating the third area function once with respect to α produces a period integral in action-angle coordinates,

Z P1 Z φ1 Z φ1 φ 1 T (P1) = dt = 2(∂αλ) dφ = dφ, C p 2 P 0 0 1 α sin(2φ) 0 − 18Letters p, q, and λ allude to momentum, position, and action quantities of Hamiltonian mechanics. 21 ∆q = 2 ←− −→ −→ = 2 p ∆ ←−

1 S(α) = 2 (2n + 1) n 0, 1, 2, 3 . ∈ { } Fig. 1.9.: A few Elliptic Curves C(α).

q φ where T (α) = 8TC (Pa) = 8TC (Pa) = 4K(α). After scaling φ and t by factors of two, we obtain a more comparable integrand, dt/dφ = Φ−1/2 with Φ = 1 α sin(φ)2. Again, the − matrices R(m) can be used to reduce the ﬁrst two α-derivatives, 1 − 1 − 3 2 3 − 1 − 3 − 5 ∂αdt = Φ 2 Φ 2 dφ, ∂ dt = Φ 2 2Φ 2 + Φ 2 dφ, −2α − α 4α2 −

As above, let w(dt) = [1, 0]T . Canonical, least coeﬃcient vectors may be written out by recursion of Hermite reduction,

T 1 3 1 1 [w(∂αdt)] = I R w(dt) = , − , − 2α − 2 · 2(1 α) 2(1 α) − − T 2 3 3 3 5 (1 3α) 1 2α [[w(∂ dt)]] = 2 I 2 R + R R w(dt) = − − , − . α 4α − 2 2 · 2 · 4(1 α)2α 2(1 α)2α − − In this next case the zero sum,

T 2 [0, 0] = w(dt) 4(1 2α)[w(∂αdt)] 4(1 α)α [[w(∂ dt)]], − − − − α

2 determines an annihilating operator. K = 1 4(1 2α)∂α 4(1 α)α∂ , The corresponding A − − − − α

22 certiﬁcate function,

Z dt Z 1 2 sin(φ)2 + α sin(φ)4 cos(φ) sin(φ) Ξφ = dφ = dφ = K E − 3/2 p A ◦ dφ 1 α sin(φ)2 1 α sin(φ)2 − − ! 2(1 2α) 3(1 α) = − − V( 3 ) + − 2V( 3 ) V( 5 ) V( 3 ) R( 5 ) w(dt), α 2 α 2 − 2 − 2 · 2 ·

φ dt dΞK allows veriﬁcation of the necessary zero sum, K = 0. It is easy to check this A ◦ dφ − dφ identity when the derivation is unavailable, misunderstood, or otherwise in doubt.

1 1 Hypergeometric annihilator K , with parameters (a, b, c) = ( , , 1), bears at least a A 2 2 superficial similarity to E. Even qualitatively, the functions E(α) and K(α) differ at their A limits, with K(α) diverging to infinity on approach to the singular point α = 1. From the geometric standpoint, the analogy is more clear between perimeter function C(α) and

area function S(α). The existence of K implies existence of another, similar annihilator, A 2 S = 1 4(1 α)α∂ , whose exact form can be calculated by solving a simple system of A − − α linear equations. By inspection, we can immediately see that S is hypergeometric with A 1 1 parameters (a, b, c) = ( , , 0), and that S admits reﬂection around α = 1/2 as an − 2 − 2 A invariant transformation. The fact that c = 0 strengthens the analogy to E and suggests A that piecewise construction of S(α) will involve no new diﬃculties. A general form for the solution is written out as,

1 1 1 1 Z 1 1 −2 2 , 2 2 , 2 2 , 2 S(α) = S1α 2F1 α S0α 2F1 α α 2F1 α dα 2 − 2 2 1 1 1 S0 2 , 2 X n = S0 + S1 + S0 α + log(α)α 2F1 α + Sn α . 8 4 2 n>1

As S(α) satisﬁes a second-order ODE, the coeﬃcients Sn with n > 1 are entirely determined

by the choice of S0 and S1. Table 1.2 lists the ﬁrst few constraints. According to reﬂection symmetry, the reversed function ) α ( S has the same formal expansion, but with integral

constants 0S and 1S . The harmonic limit toward α = 0 requires that S(α) = πα, thus

S0 = 0 and S1 = π. The opposite and strongly anharmonic limit toward α = 1 determines

0S = 4, the area of a 2 2 square, but leaves 1S undetermined. Rogue constant log(2) × 23 Table 1.2.: Constraints on S(α) P n E S(α) = cnα , n 0 A ◦ ≥ n 0 = cn = 0 0

1 32S2 4S1 3S0 − − 2 512S3 192S2 3S0 . − − . ...

n determines Sn+1

(and same for Sn nS ) →

returns to cause more trouble, and we ﬁnd by reference that 1S = 4 log(2) 3/2, − − again without a decently intelligible explanation. Proceeding pragmatically instead, we

1 match functions at α = α = 2 , sum to cutoﬀ N = 100, and calculate numerically that

1S 4.2725887222397812376689284858327063. As in the previous case of E, we do not ≈ − pw need to increase precision, and instead expand a piecewise solution S (α) only to N = 60. pw −16 Already the approximation S (α) reaches double precision of S S (α) < 10 everywhere pw A ◦ pw on the domain α [0, 1]. In subsequent analyses, we will put computable functions S (α) ∈ pw and C(α) to good use, but for now we are satisﬁed to have shown, by explicit calculation, pw exactly how solution techniques generalize from one specimen to the next.

1.6. Example Calculations

Having built both computable functions C(α) and S (α), we can also test relative convergence pw pw according to the mutual deﬁnitions of E(α) and K(α),

−16 10 > (1 2α∂α) C(α) 2∂α S (α), − ◦ pw − pw −16 10 > (1 α) + 2(1 α)α∂α (2∂α S (α)) C(α), − − ◦ pw − pw

where the inequality holds for α [0, 1]. As is reasonable to expect after Sections 4 & 5, ∈ these quality-assurance calculations reach double precision after summing only up to (α60). O 24 E, S 2π

5.105 64 . 0 4

2.5 706 .

0 α 0 α2 10 Fig. 1.10.: E(α) and S(α).

n In fact, the sums above exactly equal zero on every coeﬃcient of α when C0 2S1 = 0 − and 1C + 1S = 0C = 0S . Somewhat strangely, the geometric interpretation of identity − − 0C = 0S = 4 says that a linear distance equals an area, as does identity C0 = 2S1 = 2π between circular circumference and area. More to the point, precise veriﬁcation of interrelations between C(α) and S(α) allows us to choose just one function to investigate in detail. Derived function T (α) = 2∂ S (α) is the best to work with, because there exists pw αpw another, computationally-distinct means to calculate particular values.

19 The arithmetic-geometric mean , agm(a0, b0) = limn→∞ an = limn→∞ bn, is a recursive function of two variables, which rapidly converges on a number between successive arithmetic

1 and geometric means, an+1 = 2 (an + bn) and bn+1 = √anbn respectively. Elsewhere in [3], it is proven that T (α) agm(1 + √α, 1 √α) 2π = 0. This identity deﬁnes a numerical − − reference function T (α), with convergence dependent upon recursion depth M = n . As agm max

19 2 R π/2 2 2 2 2−1/2 Another definition is that 1/agm(x, y) = π 0 dθ x cos(θ) + y sin(θ) . 25 a first test, let us calculate for a difficult value α = 1/2 that,

1 T ( 1 ) = 1.1803405990160962260363, (N = 60) 2π pw 2 1 T ( 1 ) = 1.1803405990160962260453, (M = 7) 2π agm 2 with error beginning to show around the 20th digit (Cf. OEIS: A175574). For the same termination parameters, N = 60 and M = 7, we find that 10−20 > T (α)/T (α) 1 over |pw agm − | the domain α [0, 1). Considering that uncertainties tend to worsen as they propagate ∈ through calculations, it is not at all surprising to observe that a direct test of function values yields a tighter bound on the error due to series truncation. Ignoring more conservative tests, a sum to N = 50 already allows T (α) to reach double precision. To show off the utility of pw double-precision computable functions C(α) and S (α), we will now go through two short pw pw example calculations, only at a superficial level of detail. A problem in high-school physics asks for the magnetic field at the center of a circular

µ0I◦ loop of radius a. The answer, found by a simple Biot-Savart integral, is that B◦ = 2a zˆ, with current I◦ andz ˆ normal to the plane of the loop. A generalization of this question concerns the magnetic field at the center of a charge conducting ellipse of eccentricity e and semi-major axis a. The field is directed along the vertical, and its strength depends linearly on current strength I0 according to another not-too-difficult integral,

I I 2 I0 a dθ I0 a dl I0 C(e ) B0 = B◦ = B◦ 2 = B◦ . I◦ 2π r I◦ 2π a I◦ 2π The magnetic ﬁeld at the origin can be canceled to zero by superimposing left and right handed currents. For example, cancellation occurs between two ﬁelds B◦ and B0 when

2 B0 B◦ = B◦ B◦, or equivalently when I0/I◦ = (2π)/C(e ). Say that we choose to · − · − work with an ellipse of eccentricity e = 4/5, α = 16/25. Field cancellation requires a ratio

I0/I◦ (2π)/C(16/25) 1.23 (and we could get more digits of precision if necessary). ≈ − pw ≈ − In semi-classical quantum mechanics, another problem asks for an estimate of quantum pendulum energy eigenvalues. The period function of a simple pendulum is T (α), and its action function is the corresponding S(α). Eigenvalues αn are found by solving a qunatization 26 Table 1.3.: Semiclassical quantization of the elliptic curves (α). C 1 n S(αn) = 2 (2n + 1) Eigenvalues Percent Diﬀerence 0 0.1559223091638732 ... 0.15627 ... 0.223% 1 0.4469484490110412 ... 0.44719 ... 0.056% 2 0.7057110691134417 ... 0.70573 ... 0.003% 3 0.9212998367788911 ... 0.92011 ... 0.129%

20 1 condition such as S(αn) = 2 (2n + 1), with n = 0, 1, 2, 3. To ﬁnd the ”quantum values” −1 1 −1 αn = S 2 (2n + 1) , an inverse problem needs to be solved. Function S (s) can be found by series reversion, but this is not the most sensible approach. As we do not need an entire function, it is more expedient to simply apply a root-solving method to the zero sum

1 S (αn) (2n + 1) = 0. In so doing, we calculate the numerical values in the second column pw − 2 of Tab. 1.3, and these values are used to plot the four teal blue curves of Fig. 1.9. Alternatively, constraint α = p2 + q2 p2q2 suggests the form of a quantum mechanical − Hamiltonian matrix H, which may be written by exchanging coordinate variables p and q for their corresponding matrix representations21. We have chosen, somewhat arbitrarily, the matrix H with elements:  2  1−2i−2i +40(2i+1)π i = j  400π2   q  1 i  400π2 24 4 i = j + 4 h = h = . i,j j,i q  1 j  2 24 j = i + 4  400π 4   0 otherwise Due to non-commutation of p and q matrices, there are many diﬀerent Hamiltonian matrices, whose eigenvalues overlap S(α) within a similar range of error. Instead of dwelling on this point, we just accept the convention and move on to comparison. A few of the eigenvalues22 of H determine equally-spaced black points on the green curve of Fig. 1.10, four of which

20For more explanation, see [24] Ch. 1-3. 21Any textbook on Quantum Mechanics explains how to do so, see e.g. [37] Ch. 4,7, & 11. 22We calculate H as a 100 100 matrix with 100 eigenvalues, and select only 19 of the lowest lying. Due to duplicate values, accurate× selection requires a criterion in terms of eigenvector elements. 27 are written in the second column of Tab. 1.3. Although enumeration of roots αn to arbitrary precision shows off computational prowess, comparison with matrix eigenvalues gives apprehension as to when such efforts would actually be necessary. If the task is to approximate quantum pendulum eigenvalues to 99% accuracy, the expansion of S (α ) needs pw n far fewer than N = 60 terms. Instead of having a bikeshed digression about significant figures, let us get back to analysis of the theory itself.

1.7. Comparing Certiﬁcates

x θ Recall from Section 2 that alternative area integrals AE (P1) and AE (P1) relate to one another

x θ according the difference between certificate functions ΞA and ΞA. To continue developing the theory of certificates by an inductive process, we ask: does the derived identity ∆A(e) = (Ξθ Ξx )/e have any analog for the elliptic integrals discussed in Sections 4 & 5? And if A − A so, how do these analogs differ from ∆A(e) of the first example? The answers have curious nuances, so deserve a close look.

An incomplete arclength integral along , from initial point P0 = (p0, q0) = (√1 α, 0) E − to ﬁnal point P1 = (p1, q1) = √1 α cos(ϕ1), sin(ϕ1) is written as, − √ s Z P0 Z 1−α 2 2 p p (1 α) + αp either C (P1) = dC = − dp E E (1 α)(1 α p2) P1 p1 − − − s Z P1 Z q1 2 q q 1 αq or C (P1) = dC = − dq E E 1 q2 P0 0 − Z P1 Z ϕ1 ϕ ϕ p 2 or CE (P1) = dCE = 1 α sin(ϕ) dϕ. P0 0 −

Applying change of coordinates (cos(ϕ), sin(ϕ)) (p1 q2, q) and dϕ dq/ cos(ϕ), the → − → q ϕ dq dCE dCE dq later two of these integrals are proven equal, with dϕ dq = dϕ . Differential dϕ = cos(ϕ) q ϕ dq dq n dCE n dCE depends not on α, thus ∂α commutes with dϕ , and dϕ ∂α dq = ∂α dϕ . Consequently, integrals n q n ϕ are not merely equal. They are also identical under α-differentiation, ∂αCE (P1) = ∂αCE (P1), q ϕ q ϕ n 0. Next, certificates must equate, E C (P1) C (P1) = Ξ Ξ = 0. After ≥ A ◦ E − E E − E changing coordinates, Ξϕ Ξq = q(dq/dCq) = qp(1 q2)/(1 αq2), we can verify that E → E E − − 28 1 q1 ∆r ∆p 1 ∆p ∆q 1 p1 ∆r ∆q 2 r1 2 2 r1

( 1 p q X α + 2 1 1 dα ←− = ) p ∆ p 2

p + ∆

∆ r q 2 ∆ ( X α) ∆q −→ π φ1 (p , q ) 2 − 1 1 φ1 = r1 cos(φ1), sin(φ1) Fig. 1.11.: Tangent Geometry of curve (α). X

q q dCE dΞE E = 0 by explicit calculation (again, using a computer algebra system). A ◦ dq − dq Had we ﬁrst chosen p rather than q, the calculation would have been much worse, for dp/dϕ = √1 α sin(ϕ), and certiﬁcates do not equate, Ξp = Ξϕ . The situation is not much − − E 6 E better when comparing p and q, except that the Pythagorean theorem directly determines the hypotenuse length ∆C(α + dα) = p∆p2 + ∆q2. The series expansion23,

2 2 ! r 2 2 ∂p ∂ p ∂q ∂ q ∂p ∂q 1 2 + 2 ∆C(α + dα) = + dα + ∂α ∂α ∂α ∂α dα2 + (dα3), ∂α ∂α 2 ∂p 2 ∂q 2 O ∂α + ∂α

q p follows the variational geometry of Fig. 1.11, after deﬁning ∆C(α) = C (P1) C (P1), with E − E point P1 held independent of dα. Partial derivatives in the ﬁrst line can all be written as polynomial ratios,

∂p q2 1 ∂2p (1 q2)2 ∂q (1 q2)2 ∂2q (q2 1)3(1 + 3q2) = − , = − and = − , = − . ∂α 2 p ∂α2 − 4 p3 ∂α − 2 p2 q ∂α2 4 p4 q3

Comparison of the explicit series with a formal expansion,

1 2 2 3 ∆C(α + dα) = ∂α∆C(α) dα + ∂ ∆C(α) dα + (dα ), 2 α O

23The same expressions apply to any valid endpoint, so we omit subscript 1 on p and q variables. 29 determines ﬁrst and second variations of the arclength diﬀerence,

(1 q2)p(1 q2)2 + p2q2 ∂ ∆C(α) = − − α 2 p2 q (1 q2)2(1 + q2 5q4 + p2q4 + 3q6) and ∂2 ∆C(α) = − − . α 4 p4 q3 p(1 q2)2 + p2q2 − These data are what we need to determine the certiﬁcate diﬀerence,

2 3 2 2 2 4 6 4 4 q p (1 q ) (1 + 3q ) p (1 q 2q + 2q ) + p q E ∆C(α) = Ξ Ξ = − − − − , A ◦ E − E p2 q3 p(1 q2)2 + p2q2 − and subsequently, the missing certiﬁcate function,

4 3 5 2 p q p(1 α) + 2p (3 α)(1 α)α p (1 + 3α α ) ΞE = ΞE E ∆C(α) = −q − − − − , − A ◦ (1 α)3(1 α p2)3(1 α2) + αp2 − − − − p p dCE dΞE a truly monstrous expression! Despite gruesome details, the zero sums E = 0 A ◦ dp − dp can be checked via symbolic computation. Transferring analysis from ellipses to elliptic curves, we can guess that dependence

p q of diﬀerential dp/dq on parameter α causes area integrals SC(P1) and SC(P1) to have non-identical certiﬁcates. As in Fig. 1.11, the interior area is rectangular, with area

q p ∆S(α) = S (P1) S (P1) = qp, while the exterior triangle has area, C − C 1 ∂p ∂q dα2 ∆S(α + dα) ∆S(α) = dα2 + (dα3) = + (dα3), − −2 ∂α ∂α O −8(1 p2)(1 q2)pq O − − 2 to second order in dα. Comparison with a formal expansion determines ∂α∆S(α), and next, p2 + q2 p q S ∆S(α) = = + . A ◦ pq q p Since the curve (α) transforms invariantly by (p, q) (q, p), it is quite obvious to guess C → p q p q that either ΞS = 1/ΞS = p/q or ΞS = 1/ΞS = q/p. In fact, the first alternative is correct, p p q q dSC dΞS dSC dΞS and the two zero sums, s = 0 and s = 0, are relatively easy A ◦ dp − dp A ◦ dq − dq φ to check thereafter. A third certificate ΞS can be found by applying a similar analysis to triangular areas of Fig. 1.11. This analysis is left as an exercise for the interested reader. Having gone through a few examples of Creative Telescoping in thorough detail, the notion of a certificate function can not be as foreign. Let us now offer a summary: 30 1. If Ostrogradsky-Hermite reduction closes, invariant matrices U and V0 determine

t annihilator I , without needing to calculate the corresponding certiﬁcate Ξ . A I

t 2. When certiﬁcate ΞI can be eﬃciently calculated, it is useful for quality analysis.

dI d t Veriﬁcation of I Ξ = 0 implies that I I(α) = 0. A ◦ dt − dt I A ◦

t u 3. Two certiﬁcates ΞI and ΞI can be identical under change of coordinates, but generally they are not equal, and depend on choice of coordinates.

4. If the integral is geometric, then the certiﬁcate diﬀerence Ξt Ξu can be calculated by I − I trigonometric means, after expanding the tangent geometry in powers of dα.

The ﬁrst two observations are already well known theorems in Creative Telescoping, while the last two, if not novel, are at least lesser known. It would be interesting to generalize upon the geometric interpretation and to promote points 3 and 4 to proper theorems; however, this is outside of our present scope. We have a pragmatic perspective, and can agree that certiﬁcates are essential only at the

t level of extra rigor. Following point 1, knowledge of certiﬁcate ΞI is not necessary when constructing a function I(α), nor when evaluating I(α) at a particular value. In physics, it is often the case that a complete period I(α) is much easier to measure than any partial

R t1 integral dI/dt. A few theorists will need to derive and verify I . Once I is known t0 A A as fact, any scientist can use the well-developed theory of ordinary diﬀerential equations to construct convergence-rated approximations to I(α). The business of calculating values to I(α) then requires much less thought on the user’s end. How fortunate for them!

1.8. Prospectus

During XVIII century, Latin was a lingua franca between scientiﬁc researchers throughout Europe. According to the Euler archive, Euler wrote at least nine articles under titles starting with the word specimen, two of which are immediate to the analysis above. Around

31 the same time, Carl Linnaeus (1707-1788) began to publish Systema Naturæ (1735-1758), one of the founding documents of the modern taxonomic system in biology. Presently, the word specimen is more familiar in the Linnaen context, where it usually refers to a particular plant or animal, as collected from the wild. In other sciences, translation of specimen to ”example” is now a ubiquitous preference. We hope that the imperfect analogy will not be entirely forgotten, and that collection and analysis of specimens will continue to contribute an important part to scientific research. Ideas evolve as do plants and animals, though with entirely different constraints and rates of change. As it turns out, on planet Earth, a time sequence of accidental occurrences leads just as well to a tree of knowledge as to a tree of life. The locution that ”every new answer, leads to a few new questions” is itself a suggestion of branching structure implied by the tree-of-knowledge metaphor. Should we attempt to transfer the techniques of phylogeny—the study of evolutionary relationships—from biology to domains of pure idea? There are reasonable arguments yes and no. Before dismissing the idea as completely impossible, let us attempt to justify an evolutionary diagram such as in Fig. 1.12. This abbreviated, evolutionary flow chart attempts to trace back the existence of elliptic integrals to a momentous idea of Archimedes of Syracuse (circa 287-212 BC). His specimen of a circle bounded by two, regular 96-sided polygons is an ancestral milestone and an immense progenitor. Without it, perhaps this current work would not have come fully into existence. During antiquity, the ratio of a circle’s circumference to it’s diameter, the irrational number π, caused much frustration and eluded reasonable description. Finally after centuries of Greek thought, Circa 250 BC, the analysis of Archimedes’s Measurement of a

24 223 22 circle determined that 71 < π < 7 . This error bounded result can be reﬁned to higher precision by choosing circumscribing polygons with more than 96 sides; however, the task of doing so is prohibitively complicated. By the time of Newton, Archimedes’s technique eventually evolved to become arclength integration, so can be considered an ancestor to

24The original text is now nonextant, but a retrospective is given in [6] Ch. 13. 32 Numbers, Shapes, Contemporary Functions & Sets. Physical Science Function Theory sp. 0 + 1 + 2 < π sp. → Integrals, spp. ellipse area, circle arclength. Energy −−− −−−| ◦ sp. 6 × 24 = 96 Surfaces Series Expansions, sp. 4 arctan(1). Integral sp. Archimedes’s ◦ base case, n = 6 Periods spp. Euler’s arctan identities. Calculus Holonomic ◦ ODE’s, sp. Legendre’s identity. Functions Creative ◦ Algebraic Geometry Telescoping 2π/96 Dark/Middle Ages up to Renaissance

Roman Empire, Islamic Golden Age, European Enlightenment250BC Today −→

Fig. 1.12.: Western evolution of π (above). A drawer of closely related specimens (below).

A.1 A.2 A.3 B.1 C.1 D.1

I. Pendulum Phase Portraits E.3 B.2 C.2 D.2

II. ”Flowers from Ramanujan’s Garden”

B.3 F.1 G.1

E.1 E.2 IV. Intersection of V. Intersection of III. Quartic Strata Sphere & Ellipsoid Sphere & Goursat Surface

33 Table 1.4.: Well-Integrable Hamiltonian Energy Functions and their Period Constraints. Index Hamiltonian Function Diagnostic Period Constraint 2 2 2 2 0 I.A.1 2H = p + q p q QD 0 = T ∂α 4 α(1 α)T − − − 2 2 1 2 0 I.A.2 2H = (p + q )(1 q ) QD 0 = T ∂α 4 α(1 α)T − 4 − − 2 2 0 I.A.3 2H = p sin(q) 0 = T ∂α 4 α(1 α)T − − − 2 2 4 1 2 3 0 II.B.1 2H = p + q + ( ) 2 (3p q q ) D 0 = 2T ∂α 9 α(1 α)T 27 − − − 2 2 1 4 0 II.C.1 2H = p + q q QH 0 = 3T ∂α 16 α(1 α)T − 4 − − 2 2 4 1 3 0 II.D.1 2H = p + q ( ) 2 q H 0 = 5T ∂α 36 α(1 α)T − 27 − − 2 2 4 2 2 3 4 2 2 2 2 2 0 II.B.2 2H = p + q (p + q ) + p (p 3q ) D 0 = 8 α T ∂α 9 α(1 α )T − 27 27 − − − 2 2 1 2 2 2 2 2 2 0 II.C.2 2H = p + q (p + q ) + 2p q QD 0 = 3 α T ∂α 4 α(1 α )T − 4 − − 2 2 4 6 2 0 II.D.2 2H = p + q 27 q H 0 = 5 α T ∂α 9 α(1 α )T

34 − − − III.E.1 2H = p2 + q2 1 (p2 + q2)2 + p2q2 QD 0 = 3 α( 1) + 2T − 4 − − 2 2 1 2 2 2 1 2 2 2 0 III.E.2 2H = p + q (p + q ) + (p + q )q QD ∂α 4 α (1 α)(1 α + α)T − 4 4 − − − IV.E.3 H = a J 2 + b J 2 + c J 2 D 0 = (a + b + c 3α)T x y z − 0 (cont.) +∂α 4(a α)(b α)(c α)T √ − − − 3 2 3 2 3 2 2 2 0 V.B.3 H = J + (J 3JxJ ) (J Jz + J Jz) D 0 = 8 α T ∂α 9 α(1 α )T z 2 x − y − 2 x y − − 2 2 2 V.F.1 H = 4(JxJy) + 4(JyJz) + 4(JzJx) D 0 = 9 (4 5 α) T − 0 (cont.) ∂α 16 α(1 α)(4 3 α)T − − − V.G.1 H = J 6 5(J 2 + J 2)J 4 + 5(J 2 + J 2)2J 2 D 0 = 5 (5 21 α) T z − x y z x y z − 4 2 2 4 0 (cont.) 2(J 10J J + 5J )JxJz +∂α 4 α(1 α)(5 + 27 α)T − x − x y y − 1 4 4 4 2 2 Hotaru H = (J + J + J ) (JxJyJz) D 0 = 5(1190 13149 α + 18954 α )T 3 x y z − − 2 2 2 2 0 mirabilis (J + J + J ) ∂α 36 α(1 α)(7 27 α)(5 54 α)T − 27 x y z − − − − Diagnostic Key: Q=QuarticToODE, H=HyperellipticToODE, D=DihedralToODE. See also Appendix A. the more familiar π = R 1 (1 x2)−1/2dx. In the first generation of calculus, it was also −1 − P (−1)n possible to expand π in series, π = 4 n≥0 2n+1 , or alternatively to write π = 4 arctan(1). These definitions, though correct, are not very fit in the sense that they converge too slowly. Euler, in E74, subsequently gave numerous series approximations with improved convergence25, generalizations of the Machin formula π = 16 arctan 1 4 arctan 1 ; 5 − 239 however, even this approach was quickly out-competed by another. The Legendre identity, K(α)E( α )+K( α )E(α) K( α )K(α) = π , underpins the current best-practice for computing − 2 digits of π, so falls on another branch in the upper portion of Fig. 1.12. We choose to describe Legendre’s identity as a consequence of differential equations,

because the annihilators E, K , and S are fundamental to its deﬁnition and proof. In A A A our notation, it is better to replace E and K by S and T , which puts Legendre’s identity into a more symmetric form, S(α)T ( α ) + S( α )T (α) = 8π. First, the left-hand side is determined constant, ∂α S(α)T ( α ) + S( α )T (α) = 0. This follows from the chain rule with

T (α) = 2∂αS(α), S(α) = 2α(1 α)T (α). Second, a limit is evaluated, − lim S(α)T ( α ) + S( α )T (α) = lim α 2 log( α ) + 4 2π = 8π, α→0 α →1 × ×

by canceling the ﬁrst term to zero, and then by multiplying harmonic frequency 2π times square area 4. The proof is complete, so we can calculate an approximation of π,

2 1 1 1 1 −1 4π 2 , 2 1 2 , 2 1 π = = 4 2F1 2F1 3.14159265358979323848..., T ( 1 )S ( 1 ) 1 2 2 2 ≈ pw 2 pw 2 with only the last digit incorrect due to truncation of the series after 60 terms. Choosing α = 1/100 improves the approximation to accuracy beyond the ﬁftieth decimal place. If more precision is necessary, it is usually better to calculate T (α) and S(α) via the arithmetic-geometric mean, the Brent-Salamin algorithm, or similar [6]. The lineage of π is also a good example for showing a fallacy of the analogy between evolutionary trees. In zoology, it is never the case that a specimen of Coleoptera can identically equal a specimen of Lepidoptera, preposterous! Although both orders are

25Ed Sandifer gives a retrospective in his column, How Euler Did It, Feb. 2009. 35 arthropods and hexapods, beetles have hard shells and hidden wings, while butterflies are more delicate with obvious, exposed wings. At the finer level of classification by species, the ultimate test is whether males and females can successfully mate. Scientific ideas are gender neutral, and only ”mate” by mutual inspiration. In science, capability to mate is not at all a criterion for identity, nor for correctness. One purpose for mathematical proofs is to show that seemingly different expressions are indeed identical, or isomorphisms. Expressions equal to π are in isomorphism to one another, and this fact contradicts the basic idea of an evolutionary diagram. In summary, the descent of biological species only diverges, while the descent of ideological species diverges and converges with comparable rates. This is not a theorem, but an important observation nonetheless. When scientific ideas coalesce originally, a new field is born. Experimentation within the field leads to all sorts of novelties: examples, diagnostics, hypotheses, and eventually to theorems and proofs. This is currently happening with the field of Integral Functions. Here, Creative Telescoping algorithms are leading the way. We have already seen a few nice examples of historical importance, the well-known elliptic integrals. These are not indicative of the entire range of possibilities. According to Lairez and others, Creative Telescoping applies to any rational integral, after generalizing Ostrogradsky-Hermite reduction to its multivariate form, the Griffiths-Dwork reduction [64]. Development of broadly general methods counts as progress made toward answering big questions such as the Hodge conjecture or the Bombieri-Dwork conjecture26. Yet general progress comes at the expense of lost accessibility, and the general is not always preferable to the special. For the purposes of widespread consumption and public enlightenment, it is also worthwhile to ask: what are the most interesting examples where the simple Ostrogradsky-Hermite reduction suffices? We will continue to explore this question through a dissertation in mathematical physics. So far we have not defined the phrase ”well-integrable”, which appears in the title of this article. It is not a precise scientific term, rather a witticism about XVIII century history and

26These questions are somewhat above our pay grade, but interesting nonetheless, see also [96]. 36 language. Contemporary to Euler and Linnaeus lived Johan Sebastian Bach (1685-1750), the famous Baroque composer. Bach’s masterwork Das wohltemperierte Klavier (1722) is a source for ”præludia und Fugen durch alle Tone und Semitonia”27. The German word ”wohltemperierte” means well-tempered. It is a diagnostic term describing quality and completeness of a particular musical sample. There is no reason to worry about the exact meaning of either term ”well-tempered” or ”well-integrable”. Instead, we give the examples of Fig. 1.12 and Tab. 1.4, and allow the observer to form his or her own opinions. Isn’t it easy to wonder if the letters A-G have something to do with key signatures in western music? In good humor, the answer could be yes, in sober scientific explanation, no. Let us briefly explain one difference between musical and scientific indexing schemes. According to the order of sharps, the musical keys have a second arrangement, FCGDAEB, which is more telling than the alphabetic arrangement. In this arrangement, the second ”Crystal Clear” key of C is usually thought of as a starting place. Indeed, Bach chooses C Major as they key of the first prelude and fugue in Das wohltemperierte Klavier. In our presentation of the few well-integrable geometries, the initial example, chosen as A, is followed immediately by similar examples B,C, and D, subsequently generalized by the examples of E28. Only then is it possible for us to exceed the learning curve and complete skills development by transferring analysis to examples F and G. Unlike the musical system, the letter indices are neither finite nor cyclic. There could have been an VI.H.1 (not depicted in Fig. 1.12) for the mysterious geometry Hotaru mirabilis. After that, we might even have listed a few more in Tab. 1.4, but simply ran out of room on the page. All jokes aside, our chosen examples A-G must belong together. Many identities between them support appropriateness of the arrangement, and relative completeness can be explored through combinatorial analysis. The most obvious similarity is that, according to the diagnostic algorithms HyperellipticToODE and DihedralToODE, all derived period

27Impressively, AndrásSchiff plays the entire Book I by memory on youtube. This is good way to experience music that inspired many subsequent works, including Hofstatder’s Godel, Escher, Bach. 28The letters A-D were originally used by Almkvist and Zudilin [5] p. 20. Letter E is chosen as in Euler. Euler was among the first to study rigid body rotations, leading eventually to IV.E.3. 37 constraints are second-order ordinary differential equations. Examples A-D are all of the hypergeometric type, while examples E,F, and G are Heun equations, i.e. they have exactly four regular singular points. The cohesion is even stronger, because annihilators of the

2 respective period functions all have the form T = a0(α) + ∂αa2(α) ∂α + a2(α)∂ , with A α just two polynomial coefficients a0(α) and a2(α). This allows factoring of the corresponding ODE, as in Tab. 1.4. Previously, Fritz Beukers and Don Zagier explored this form in connection with Apéry’sproof of the irrationality of ζ(3) [102]. Zagier also found the set A-D as complete, but the finite scope of his massive search precluded the possibility of finding any of the subsequent examples from Tab. 1.4. All of the geometries listed involve ”integrality miracles”, most notably Hotaru mirabilis29. Yet we must doubt our own bias, and act as potential naysayers to ourselves by asking: Why the results of Fig. 1.12 and Tab. 1.4? Granted, some importance derives from relation to leading mathematical theories, such as the KZ-Periods, Creative Telescoping, Holonomic functions, lattice walk Combinatorics, Cohomology, etc. These theories are only part of the picture, high and far away from the concerns of most scientists. Ultimately our answer why derives from leading physical theories, where integral periods are also laboratory observables. This is true throughout classical Hamiltonian mechanics, and more specifically in the subsequent theory of semi-classical quantum mechanics. Up until now it has not been very well understood that it is possible to calculate a semi-classical level spectrum by root-solving the solution to an ODE. This is a central ”theorem” that will arise out of the subsequent exposition. By developing the examples E, F, and G, we can improve upon the rigor of the original analysis by Harter et al. (cf. [46, 48, 50]). The theory of Rotational Energy Surfaces can then grow in new directions parallel to those of pure mathematics. The entire course will require hard work and effort on the readers part, but the payoff is immense. Not only are the results useful in physics calculations, they are also beautiful in their own right!

29This east-meets-west name commemorates discovery ”by the light of the fireflies”, and that the period ODE begets integrality miracles around all four finite-valued, regular singular points. 38 The reader may complain or protest, how dare we present results on Hamiltonian mechanics without even defining terms? This is a fine objection, but in the end, not one that will successfully dispute validity. According to the supplemental materials described in Appendix A, the entire Table IV is automatically proven correct in under five seconds. In this exsanguine process, a ”Hamiltonian function” is nothing more special than an input to a diagnostic algorithm whose output is an ordinary differential equation—not too exciting. To answer the objection, we do need to explain more about what a Hamiltonian function is, and about how to build Hamiltonian mechanics until it is capable of describing rigid body and semi-rigid body rotations. This is not possible in a prelude, nor in a fugue. It will take the time and space of a few more dissertation chapters. Only then will the diagnostic algorithms, and the resulting alphanumeric classification begin to make real sense. More importantly, only after these next few chapters can we begin to understand, appreciate, and improve Harter’s theory of Rotational Energy Surfaces. There will be a few mathematical loose ends, but these too can be appreciated for what they are worth. We look forward to a conclusion where the small mystery of Hotaru mirabilis will flash and blink and drift away into the darkness around. Without further ado, let the voices of science intertwine. . .

Acknowledgements

The Online Encyclopedia of Integer Sequences was a source of inspiration, see for example: A002894, A113424, A000897, A006480, A318245, A318495. Interest and helpful comments of OEIS volunteer editors, especially through [seqfan] mailing list, helped immensely in formulating this analysis. Related discussions on [mathfun] mailing list also contributed positively, with motivation from Bill Gosper, and references [33] and [68] were given by Mike Stay and Alan Wechsler, respectively. Don Zagier suggested Ref. [102]. Calculations and drawings are the author’s own work (usually with the aid of Mathematica [54]).

39 2. An Alternative Theory of Simple Pendulum Libration

2 2 Integration along the transcendental Hamiltonian function α = 2Hϑ(p, q) = p + sin(q) determines the time dependence of simple pendulum libration when α (0, 1). By ∈ equivalence via canonical transformation, the same can be said for the algebraic Hamiltonian function,

2 2 1 2 α = 2Hϕ(p, q) = p + q 1 q , − 4 a close relative to Harold Edwards’s normal form for elliptic curves. Combining real and complex transformation theory with Edwards’s theory of elliptic curves and elliptic functions, we derive an exact solution of the simple pendulum’s librational equations of motion. Theory is tested by an experiment where a ﬁdget spinner pendulum undergoes libration. In a ﬁrst analysis, we characterize the functional form of period vs. energy data by extracting a shape parameter to acceptable accuracy and precision.

2.1. History and Introduction

The universal law of gravitation1 guarantees an analogy between the libration of a pendulum around its fixed axis and the orbital motion of a planet around its central star. The entire analogy goes deeper than to simply describe a planet and a pendulum bob as massive objects affected by gravitational forces. In both cases, the equations of motion have oscillating solutions, which return to their initial conditions after a fixed interval of time. In both cases, the recurrence periods depend on an amplitude parameter that enters through the initial conditions. Here the analogy reveals its great fallacy. The two period functions do not share very many similarities, nor are they equally difficult to solve or to measure. The discovery and resolution of this dichotomy plays across many interesting chapters from the history of science, and leads into the subject matter of this current work. About a century before the advent of integral calculus, Galileo (1564-1642) made the simple

1For an introduction, try Feynman Lectures on Physics, Vol. I, Ch. 7 [30], or refer back to [59]. 40 T

α 0 1 Fig. 2.1.: Two period functions.

observation that two pendulums of the same characteristic length will undergo isochronous oscillations. The observation does not hold true outside the small angle limit, as Galileo may very well have known2. A counterpoint to this story occurs in the researches of Kepler (1571-1630), who ﬁrst discovered the three laws of planetary motion. The third law asserts that the square of a planet’s yearly period is proportional to the cube of the orbital ellipse’s semi-major axis length. In a sense, Kepler succeeded where Galileo could not. He was able to correctly ascertain the functional dependence of an oscillation period. Meanwhile, signiﬁcant obstacles in theory and experiment stood between renaissance scientists and a better understanding of pendulum dynamics. Though the planets can not be controlled in a laboratory as can a pendulum bob, the planetary period function is sometimes easier to measure than that of a simple pendulum3. Using only XVI century implements, an astronomer can chart the day-to-day motion of a

2As Bill Gosper speculates in an email from Sunday July 7, 2019, 05:33:49 UTC. 3For a popular discussion of Galileo, Kepler, and early measurements, try Infinite Powers [93]. 41 planet at a resolution of tens or hundreds of data points per yearly cycle. Without digital acquisition of data, a similar sampling of pendulum motion at 10 to 100 Hertz is practically impossible. Dissipative losses introduce uncertainty to measured values and compound upon difficulties due to short timescale. Even if renaissance experimental scientists could have accurately measured changes of the pendulum’s period on the order of a few percents, the analytic task of extracting a functional form would have been an impossible prospect before the theoretical advances of Euler (1707-1783). The period function in question is not algebraic, nor is it immediately easy to characterize by fitting only a few free parameters. Among hundreds of other topics, Euler initiated the general study of elliptic integrals and hypergeometric series, and subsequently solved exactly for the simple pendulum’s amplitude dependence4. Careful examination of publication records reveals yet another example of Arnold’s principle that ”discoveries are rarely attributed to the correct person”. Euler could already understand the hypergeometric function 2F1(a, b; c; z) in terms of its characteristic differential equation,

2 a b F c (a + b + 1)z ∂zF z(1 z)∂ F = 0, − − − − z yet most authors credit the later Gauss (1777-1855), who gave a more extensive treatment and standardized notation, decades after Euler’s exploratory work [26]. In any case, the hypergeometric function eventually came to be seen as a crown accomplishment of classical function theory and a sine qua non of period analysis [61]. Up to a scale degree of freedom or an initial value, the two ordinary diﬀerential equations,

3 T 2 α ∂αT = 0 vs. T ∂α(4 α (1 α) ∂αT ) = 0, − − − deﬁne the periods of planetary and pendulum motion respectively, as seen in Figure 2.1. Integers 2 and 3 appearing in the ﬁrst equation are the exponents already known to Kepler. Integer 4 appearing in the second equation is not so easy to explain. Choosing hypergeometric

4Cf. Euler Archive [28]: E028, E366, E503, E710. 42 a 1 s = 2, 3, 4, 6, 9, 14, 22, 45.

1 2

b 1 2 1 Fig. 2.2.: Mapping between parameters. parameters (a, b, c) = (1/s, (s 1)/s, 1), the general equation reduces to a factored form, −

2 (s 1)F ∂z s z (1 z) ∂zF = 0. − − −

Usually the line a + b = 1 intersects hyperbola a b = (s 1)/s2 at two distinct points, − while preserving reflection symmetry across the line a = b. Choosing s = 2, line a + b = 1 lays tangent to hyperbola a b = 1/4 at the symmetry point a = b = 1/2, as in Fig. 2.2. This choice also recovers the pendulum’s characteristic differential equation, thus the appearance of integer 4 acquires a special meaning through the hypergeometric theory. The decomposition of 4 to (a, b, c) = (1/2, 1/2, 1) produces all the necessary parameters to construct a series solution of the simple pendulum’s period (Cf. [35] Ch. 5 or [59] Section 5). Euler and Gauss developed a fine characterization of the gross dynamics of a simple pendulum, but left exact solution of the time parameterization problem mostly an open question for the next few generations of researchers. The answering involved an honor roll of European patriarchs—Legendre, Riemann, Jacobi, Weierstrass and many others. Especially through the efforts of Abel (1802-1829), a central theme emerged that the time parameter can also take on complex values5. Subsequently during XX century the pendulum’s exact solution

5For another closely related historical account, try What is the Genus? [74]. 43 Fig. 2.3.: Sommerfeld’s phase plane geometries. (Public Domain via archive.org)

in terms of standard, doubly-periodic elliptic functions became a lauded final product of the classical era [99, 100]. The XX century also saw the advent of Quantum Mechanics. During this time period, precise spectroscopic measurement of atoms and molecules confounded preexisting theories; however, physicists did ultimately discover that classical oscillations have somewhat odd counterparts in the quantum regime. Initial attempts to describe quantum oscillations built upon the concept of phase space. The mathematical prehistory of phase space goes back to Poincaré(1854-1912), but the work of Ehrenfest (1880-1933) marks the first occurrence of the the Deutsch word Phasenraum [70]. Thereafter Sommerfeld (1868-1951) included early depictions of two equivalent phase plane geometries in his famous text Atombau Und Spektrallinien [90]. To the uninitiate, the two drawings reproduced here in Fig. 2.3 are only curious works of abstract art. It is not immediately apparent that they describe harmonic oscillation, nor that they relate by a special coordinate transformation, nor that they can be deformed to account for non-linearity. These few basic facts of Hamiltonian Mechanics are now widely understood, and they are still important today6. Modern texts follow and expound upon the figure drawing of Sommerfeld and others, making Hamiltonian mechanics an attractive and delightful subject, especially for visual learners and abstract free-thinkers. The pendulum phase portrait, here Fig. 2.4, is an

6Especially in the development of The Semiclassical Way [52]. 44 p

Fig. 2.4.: Pendulum Phase Portrait.

iconic standard. Both popular and specialist accounts7 usually include such a figure when developing the visual language. The more demanding references typically ask the student not only to recreate the figure drawing, but also to measure its dimensions in terms of a period integral. Strangely enough, the typical solution does not usually involve much geometry, instead shows that a pendulum’s dimensionless falling/rising velocityw ˙ along the vertical coordinate w satisfies an algebraic constraint,w ˙ 2 = 1 w(1 w)(1 αw). Change of variables 4 − − by w = sin(φ)2 then yields the textbook form,

Z 1 dw I dφ T (α) = 2 = . p p 2 0 w(1 w)(1 αw) 1 α sin(φ) − − − Avoiding mimicry, we will use geometric methods to ﬁnd, and to prove valid, a canonical Hamiltonian formulation where q/p = tan(φ) and φ˙ = p1 α sin(φ)2. Discovery (or − 2 2 1 2 rediscovery) of the algebraic Hamiltonian function 2Hϕ(p, q) = (p + q )(1 q ) leads − 4 not only to quick derivation of the period integral, but also to an exact solution of the time parameterization problem via the Harold Edwards theory of elliptic curves and functions. The article ”A normal form for elliptic curves” testiﬁes to Abel’s inspirational genius, and

7For example: [93] Chapter 11; [92] Chapter 6; [51] Unit 2, Chapter 7, [65] Chapter III, etc. 45 stands as a paragon work of alternative perspective [27]. History leads Edwards and his readers to a thoughtful and self-consistent revision of elliptic function theory. Simplification by symmetry is an important theme in this work of pure mathematics, but the results are not solely the product or possession of a leisure class. Interdisciplinary applications are part of the history and its followings. In cryptography, the simplified addition rule helps to optimize implementations of the widely-used Diffie-Hellman key-exchange protocol [14]. Thus computer scientists have been among the first to accept and utilize the alternative paradigm. Edwards’s normal form also presents physicists with an opportunity to break free from the confines of disciplinary boundaries and standard formulae8. Perhaps the future will have a better role for ”Hamilton-Abel theory” than that of a clever joke or an idle dream. As a first step to actualizing H.A. theory, we will develop a synthesis between real-valued Hamiltonian mechanics and the sort of complex-valued calculus present in Harold Edwards’s original masterpiece. Although our primary interest is theoretical, part of the draw to simple pendulum analysis is the simplicity of the experimental set up. Experimental determination of the harmonic period is ubiquitous in undergrad laboratories, while the period’s amplitude dependence is often ignored. Now that high-speed recording devices are readily available, it is easy to obtain data at sufficient precision to ascertain the functional form of T (α). In practice, this requires us to introduce a shape parameter and a trial function T (α, ) such that T (α, 1) = T (α). In the final section, we manage to obtain high-precision data and to extract = 0.98 0.03, ± thus leading the way to a new standard of experimental analysis.

2.2. Preliminary Analysis

The simple pendulum consists of a massive bob attached by a solid rod, assumed massless, to an axle as in Fig. 2.5. Gravity acts on the bob with vertical force mg, and the attachment applies a response force. The rod feels extensive and compressive stresses, but is assumed

8As hinted during ”ECCHacks” @ 31C3, Chaos Communication Congress 2014 [15]. 46 z

s g θ0

θ a0

m x

Fig. 2.5.: Simple Pendulum Geometry.

to respond with zero strain. As time elapses the bob swings and undergoes periodic motion along a circular trajectory of radius l. In librational motion the signs of angular coordinate θ and angular velocity θ˙ alternate while the pendulum reaches extremum deﬂection at regular intervals throughout the experiment. The time of one complete oscillation is called the period and denoted by symbol T . In the absence of frictional damping a time series starting from

time t = 0 at initial angle θ0 would have that θ(t) = θ0 whenever t/T N. More realistically, ∈ one period Tn separates successive maxima θn and θn+1, with a non-zero frictional loss

δθn = θn+1 θn > 0. Only in the limit where δθn approaches zero, period Tn = Tn+1 − becomes an exactly integrable observable. We shall work out the theory assuming that

δθn = 0 regardless of n, while postponing worries about fidelity to a followup article. The simple design of Fig. 2.5 involves a few free parameters, collected in Table 2.1. None of the initial parameters have dimension [ T ] for time, but [ T ] does occur as a factor in the dimensions of g. Without loss of generality, the [ L ][ M ][ T ] dimensional system allows us to fix three scale degrees of freedom, i.e. to choose the base units of length, mass, and time. The obvious choice l = m = 1 scales dimensions [ L ] and [ M ]. Another sensible choice, that g = 1, forces l/g = 1, so also sets the [ T ]-scale. By intuition, we can easily guess period T proportional to fixed time pl/g = 1. Such a guess pays no regard to the

47 Table 2.1.: More Parameters. Symbol Dimension l [ L ]

a0 [ L ] g [ L ][ T ]−2 m [ M ] (Length, Mass, Time )

dimensionless amplitude parameter,

1 2 α = a0/l = 1 cos(θ0) = sin(θ0/2) . 2 −

Parameter α presents a diﬃculty to dimensional analysis by allowing that T is a function and not just a single number. For any integer n, the quantity pl/g αn has time dimension, p as does any quantity T 2π l/g Q[[α]]. An Ansatz for the period function, ∈ s l X T (α) = 2π c αn, g n n≥0

includes infinitely many undetermined coefficients cn Q. These coefficients are not ∈ constrained by dimensional analysis. The task of determining them calls for a stronger approach, one predicated upon a combination of physical principle and integral calculus. Let the simple pendulum move in the xz plane along the arc of a unit circle. Choosing the center at (x, z) = (0, 1) requires that x2 + (z 1)2 = 1, and that x x˙ = (1 z)z ˙. By − − these two equations, the bivariate expression for conservation of energy, 2(2α z) =x ˙ 2 +z ˙2, − reduces to a univariate form,z ˙2 = 2z(2α z)(2 z). Re-scaled variable w = z/(2 α) enables − − neat expression of the rising/falling velocity, but even greater simplicity follows from the choice of w = sin(φ)2. The period integral then takes the form given in the introduction. Quarter-period integral

π Z 2 dφ K(α) = T (α)/4 = , p 2 0 1 α sin(φ) − 48 is such a famous standard that it has a long-winded name, the complete elliptic integral of the ﬁrst kind [65]. By formally verifying that T (α) = 4K(α), part (a) of the typical textbook exercise is already completed. More challenging part (b) asks for a closed form for

9 the coefficients cn. Term-by-term integration of the series expansion yields a closed form for each cn, 2π 2π 2 I dφ X 1 2n I X 1 2n T (α) = = αn sin(φ)2n = 2π αn. p1 α sin(φ)2 4n n 16n n − 0 0 1 2n2 If both parts (a) and (b) are too easy, part (c) asks for proof that cn = 16n n 2 2 satisfies (n + 1) cn+1 = (n + 1/2) cn, and consequently that K(α) satisfies a particular hypergeometric differential equation10. Unfortunately textbook answers to parts (a), (b), and (c), do not contribute at all to intuition for the mysterious variable φ. They are not so much answers, but instead a distraction from deeper inquisition. Why does change of variables from z to φ work so well? Does simplicity indicate hidden meaning? Is φ actually the angle of some particular geometric figure? Recall that each libration cycle involves an alternating pattern of extrema,

max(θ˙) K max(θ) K min(θ˙) K min(θ) K max(θ˙), −−−−→ −−−−→ −−−−→ −−−−→ separated by equal time intervals of length K(α). This notation suggests that the abstraction φ relates somehow to the phase between hanging angle θ and its angular velocity θ˙.A simple hypothesis makes φ the angle of a polar coordinate system where θ = r sin(φ) and θ˙ = r cos(φ). If correct, the hypothesis should enable another derivation of the exact same integrand dφ/φ˙. Gravitational force mg = 1 downward along the vertical results in a partial force along the tangent of motion, which aﬀects an angular acceleration, and by Newton’s laws, θ¨ = sin(θ). The phase angular velocity, − d θ˙2 d θ θ˙2 θθ¨ φ˙ = cos(φ)2 tan(φ) = = − , dt θ2 + θ˙2 dt θ˙ θ2 + θ˙2 sin(φ) X ( 1)n = cos(φ)2 + sin r sin(φ) = 1 + − r2n sin(φ)2n+2, r (2n + 1)! n>0

9 −1/2 P 2n n 1 H 2n 2n Using that (1 4Φ) = n Φ and that 2π (2 sin(φ)) = n . Cf. OEIS [89], A000984. 10We leave this− as a worthwhile exercise for the reader. Cf. OEIS [89], A002894. 49 looks hopelessly complicated. It is about to get even worse. Conservation of energy, 4α = θ˙2 + 4 sin(θ/2)2, determines r as a function of sin(φ) and α by reversion of the series,

n X ( 1) 2n+2 4α = r2 + 2 − r sin(φ) . (2n + 2)! n>0

Fortunately, the current argument depends only on the linear variation of T (α). The inverse series takes a form r2 = 4α + (α2), and substitution to φ˙ partially solves T (α), O I dφ I 2 T (α) = = dφ 1 + α sin(φ)4 + (α2) . φ˙ 3 O

1 H 4 Integral identity 2π dφ sin(φ) = 3/8 ensures the correct value c1 = 1/4. More importantly, the expansion shows by contradiction that tan(φ) = θ/θ˙ because the linear term of T (α) is an 6 integral of sin(φ)4 rather than sin(φ)2. The false hypothesis is not entirely a loss. Similarity between alternative calculations of T (α) supports the idea that φ could be a phase angle. To ﬁnd and prove the correct geometry, we will next give a short development of Hamiltonian mechanics, including a few basic facts from transformation theory.

2.3. Phase Plane Geometry

The phase plane is a two-dimensional, Euclidean vector space spanned by Cartesian (p, q) variables. These variables measure the state of a test mass as it undergoes classical motion along one dimension. A choice of coordinates involves inﬁnitely many hidden degrees of freedom, so it is usually not true that p stands for momentum and q for position. Hamiltonian mechanics reserves the letters p and q for those canonical coordinates, which are deﬁned to satisfy Hamilton’s equations of motion,

d ∂H ∂H p, q = p,˙ q˙ = , . dt − ∂q ∂p

Additional variables H and t stand for the Hamiltonian energy function and the special time parameter, respectively. Again, we will assume 2H(p, q) = α withα ˙ = 0, thus the Hamiltonian H uniquely determines the conserved total energy at any state-point (p, q). A

50 α

q p

Fig. 2.6.: A Circular Paraboloid ◦. H

three-dimensional visualization of function H(p, q) superposes a surface,

3 = (p, q, α) R : α = 2 H(p, q) , H { ∈ } above the phase plane. Level sets of surface project to phase curves, H

2 (α) = (p, q) R : α = 2 H(p, q), (p, ˙ q˙) = (0, 0) . C { ∈ 6 }

The tangent expansion of phase curve (α) at energy α, C ∂H ∂H dα = 0 = dp + dq =q ˙ dp p˙ dq, ∂p ∂q −

requires the existence of an invariant diﬀerential, dt = dp/p˙ = dq/q˙, which solves the linear constraint equation. The integral t = R dt determines the time parameter t of (α) and, by C integral inversion, solutions q(t) and p(t) to the equations of motion. At critical points where (p, ˙ q˙) = (0, 0), a tangent does not exist. Instead, the solution, p(t) = p(0) and q(t) = q(0), follows from the constant value theorem. In a case-by-case exposition of Hamiltonian mechanics, harmonic oscillation usually shows within the ﬁrst few examples. The Hamiltonian,

2 2 2H0(p, q) = κp p + κq q , κp > 0, κq > 0 51 describes a paraboloid 0. The phase curves 0(α) are concentric ellipses when α > 0. At H C the point (p, q) = (0, 0), the system reaches a stable minimum with α = 0, positive principal

curvatures, κp and κq, and harmonic frequency ω = √κp κq. Up to an initial condition φ0, trigonometric functions,

q q p(t) = α/κp cos(ω t + φ0) and q(t) = α/κq sin(ω t + φ0),

solve Hamilton’s equations, (p, ˙ q˙) = ( κq q, κp p). A guess-and-check strategy works ﬁne − when the derivatives of sine and cosine are already known. Repeating the solution in polar coordinates helps to develop better insight.

Choosing dimensions κp = κq = ω puts paraboloid 0 into a most symmetrical, circular H form, as in Fig. 2.6. In polar coordinates, p, q = r cos(φ), r sin(φ), the Hamiltonian

2 2H◦(r) = ωr depends only the radial coordinate and leaves the phase angle unconstrained. Conservation of energy sets radius r = pα/ω. Again from tan(φ) = q/p it follows that

˙ 2 2 2 2 φ = (pq˙ pq˙ )/r = ω(p + q )/r = ω and that φ = ωt + φ0. This calculation veriﬁes the − time parameterization, but more importantly allows proof, ∂rH◦(r) = ω, that r and φ do 6 not make a pair of canonical coordinates. Instead, the canonical choice is λ = r2/2, and then Hamilton’s equations work perfectly well,

˙ ˙ φ = ∂λH◦(λ) = ω and λ = ∂φH◦(λ) = 0. −

This observation unlocks the secret of Sommerfeld’s second drawing. It is a plain picture of the harmonic oscillator phase curves in action-angle coordinates (p, q) = (λ, φ). Action-angle coordinates are an extremely useful tool, and worthy of a proper introduction. When stated seriously, the changing of a phase ellipse into circular form gives a seminal example of canonical transformation theory. Squeeze transformation of the phase plane,

(pi, qi) (pf , qf ) = (pi k, qi/k), recovers H◦(pf , qf ) from the earlier H0(pi, qi), when → 1/4 k = (κp/κq) . Hamilton’s equations remain invariant,

∂H ∂H ∂H ∂H p˙i, q˙i = , p˙f /k, q˙f k = /k, k , − ∂qi ∂pi −→ − ∂qf ∂pf 52 as the extra factors k cancel. Similarly, factors of k cancel after transforming the area two-form, dp dq. This is no coincidence. When coordinate q holds constant, the tangent ∧ geometry requires that dp/dα = (2q ˙)−1, and similar for dq/dα with p constant. After R applying Stokes’s theorem, the time integrand obtains a profound form, dt = 2∂α dp dq. H ∧ We can now ask a fundamental question: which coordinate transformations leave dt invariant, or equivalently, which transformations leave Hamilton’s equations invariant? Under a general transformation (pi, qi) pf , qf the area form, → ∂pf ∂pf ∂qf ∂qf dpf dqf = dpi + dqi dpi + dqi = det(J) dpi dqi, ∧ ∂pi ∂qi ∧ ∂pi ∂qi ∧ scales according to the Jacobian matrix and its determinant,   ∂pf ∂pf  ∂pi ∂qi  ∂pf ∂qf ∂pf ∂qf J =   and det(J) = .   ∂pi ∂qi − ∂qi ∂pi ∂qf ∂qf ∂pi ∂qi After characterizing the inverse transform (pf , qf ) pi, qi with an analogous matrix →

J = 1/J T , action on the tangent vectors,

(dpf , dqf ) = J (dpi, dqi) and (dpi, dqi) = J (dpf , dqf ). · ·

ensures a product to identity, J J = I. Hamilton’s equations transform accordingly,

· J ∂H ∂H T T ∂H ∂H p˙f , q˙f = , J p˙i, q˙i = P P , ,

− ∂qf ∂pf −→ · · · · − ∂qi ∂pi

J J

with permutation matrix P such that P T TJ P = det( ) −1 = J/ det(J). When det(J) = 1, · · the area form and Hamilton’s equations remain invariant11. The change of coordinates is then said to be a canonical transformation. For example, the pair of Jacobians,    

p p p q J cos(q )/ 2p 2p sin(q )  i i   f f f f  J =   and =  −  ,     2 2 2 2 p p qi/(p + q ) pi/(p + q ) sin(qf )/ 2pf 2pf cos(qf )

− i i i i J satsify det(J) = det( ) = 1, thus action-angle coordinates (pf , qf = (λ, φ) are proven

12 canonical relative to position-momentum coordinates (pi, qi) = (p, q). 11This proof is adapted from [75]. 12This fact also follows quite obviously from the simple calculation, dλ dφ = rdr dφ, because the right hand side reproduces the well-known area form of a polar coordinate∧ system. ∧ 53 t

T −→

T −→ q p

Fig. 2.7.: A Helicoid Flow ◦. F

Jacobian matrices simplify validation of coordinate transformations, but they are no substitute for Hamilton’s equations. The integral form,

Z t ZZ 0 t = dt = 2∂α dp dq = 2∂αS(α, p, q), 0 H ∧ tells more about time, that its parameter t depends on the area S(α, p, q) interior to curve (α), between a valid initial and ﬁnal condition, (p0, q0) and (p, q) respectively. The C Hamiltonian ﬂow,

3 = p, q, t R : t = 2∂αS(α, p, q) , F ∈ equates time evolution with a solid geometry in phase-space-time (Deutsch: phasenraumzeit).

For harmonic oscillation, we can choose initial conditions such that 2S◦(α, λ, φ) = (φ/ω) α, and then the ﬂow goes along a helicoid spiral over circular phase curves. In Fig. 2.7, the helicoid ﬂow repeats after a vertical time-translation of T◦ = 2π/ω, because any closed curve

2 ◦(α) bounds an entire area, S◦(α) = πr = (π/ω)α. Amplitude independence of period T◦ C characterizes harmonic oscillation. More generally, anharmonic oscillation along curve (α) C returns to initial condition (p0, q0) after a period T (α), which characteristically must vary with energy α. Usually amplitude dependence cannot be avoided, and then the harmonic 54 equations of motion are, at best, valid and useful only as an approximation in the inﬁnitesimal limit. The most interesting phase plane geometries involve a spatial admixture of qualitatively diﬀerent critical points. In local coordinates (p, q) around a critical point at (p, q) = (0, 0),

2 2 assume the Hamiltonian becomes approximately quadratic, 2H(p, q) κpp + κqq . Then ≈ the sign of κpκq distinguishes between circular points with κpκq > 0 and hyperbolic points with κpκq < 0. We have already seen that circular points imply local harmonic oscillation, but have yet to encounter hyperbolic points or the separatrix curves they pair with. Choosing

2 2 coordinates where 2He◦(p, q) = p q , hyperbolic trigonometric functions, p(t) = √α cosh(t) − and q(t) = √α sinh(t) solve Hamilton’s equations, (p, ˙ q˙) = (q, p). Proof does not diﬀer signiﬁcantly from the earlier case of Harmonic oscillation. In fact, this similarity between

13 H◦ and He◦ is the ﬁrst sign of an even deeper transformation theory .

Around a hyperbolic point, the Hamiltonian factors, 2He◦(p, q) = (p + q)(p q). Lines − determined by p + q = 0 and p q = 0 intersect at the hyperbolic point. A separatrix curve, − with α = 0, extends from those two lines outwards to the wider area of the phase plane. When the Hamiltonian contains terms higher than quadratic, a line leaving the hyperbolic point can possibly change direction. Then the separatrix segment either goes off to infinity, or it may return along another direction to the initial hyperbolic point, or it may even approach a second, distinct hyperbolic point. Taking a union over disjoint segments, the separatrix curve is so named because, globally, it separates the phase plane into qualitatively different, non-intersecting subsets. For example, the red separatrix curve of Fig. 2.4 separates the phase portrait into regions where the pendulum rotates either clockwise or counterclockwise, and another central region where libration occurs. For our purposes, an oscillation disk is defined as a topological disk within the phase plane,

bounded by a separatrix at energy α1, and containing exactly one critical point, a circular

point at energy α0. This domain also contains anharmonic oscillation within the energy range

13The general case, H He, is discussed thoroughly in Section 4. Incorporating Complex Time. → 55 2 2 2 2 2 2 1 2 2 2 2H1 = p + q p q 2Hϕ = (p + q )(1 4 q ) 2Hϑ = p + sin(q) 2 2− − 2 φ˙ = 1 2 p q φ˙ = 1 1 q2 φ˙ = p +q cos(q) sin(q) − p2+q2 − 2 p2+q2 φ˙ = 0 1 { } Fig. 2.8.: A Few Oscillation Disk Heatmaps, Colored by φ˙.

α (α0, α1). Level curves around the circular point gradually deform away from a circular ∈ shape, but must retain loop topology. When a curve accumulates non-constant variation of curvature, the phase angular velocity φ˙ measures change of shape, as in Fig. 2.8. Most obviously, the heatmaps show slowing around the hyperbolic points. On a general oscillation disk, the limit α α1 causes at least one small interval of the curve (α) to pinch into → C a corner near a hyperbolic point. Such an extreme deformation forces φ˙ to approach zero locally, and the integral period function diverges. The pendulum phase portrait of Fig. 2.4 includes an oscillation disk at low energy, which appears again in the right of Fig. 2.8. Choosing canonical coordinates (p, q) = (θ,˙ θ)/2, the pendulum’s Hamiltonian energy function is written,

2 2 α = 2Hϑ(p, q) = p + sin(q) ,

over energy domain 0 < α < 1. Within one period, π 2q π, a circular point occurs ≤ ≤ at (q, p) = (0, 0), and hyperbolic points occur at (p, q) = (0, π/2). The period function ± must increase from its harmonic limit at α = 0 to an infinite divergence at α = 1. Limiting analysis recapitulates what we should already know from preliminary analysis and laboratory experiments. At small amplitudes, the pendulum oscillates harmonically around the stable minimum. Larger amplitude oscillations eventually reach an inverted configuration where the force of gravity has only a small component along the direction of motion. Thus slowing 56 occurs on approach to the unstable equilibrium point. The mere existence of singular divergences suggests that ordinary differential equations could be a useful tool for rigorously defining the pendulum’s period function. At first we will prefer an easier, more direct analysis. Up to a second power of the action variable λ, the pendulum’s Hamiltonian may be approximated as α/2 = H(λ, φ) = λ 2 λ2 sin(φ)4. − 3 q Quadratic root solving yields twice the action 2λ = 3 1 1 1 4 α sin(φ)4 , and by 2 sin(φ)4 − − 3 the chain rule, I dφ I I dφ 1 35 T (α) = = 2 dφ ∂ λ = = 2π 1 + α + α2 + ... . ˙ α q φ 1 4 α sin(φ)4 4 192 − 3 This is a nice and easy trick! It gets the linear term correct, without any need for series reversion. However, the sine function is bounded by 1, so the denominator goes to zero ± p when α = 3/4, and the critical points fall short at q = 3/2 < π/2. After c0 and c1, | | 14 all coefficients cn with n > 1 are apparently overestimates . Instead, let us introduce an arbitrary trigonometric polynomial Φ and replace (4/3) sin(φ)4 Φ. When Φ = sin(φ)2, → the root solving procedure reproduces the correct period function. As Φ multiplies λ2 in the Hamiltonian, any valid perturbing term needs Φ to be a homogeneous quartic polynomial in the variables P = cos(φ), Q = sin(φ). The quadratic choice Φ = Q2 obviously does not satisfy the validity constraint, but a workaround is available via the Pythagorean theorem, P 2 +Q2 = 1. Multiplication of Φ by the algebraic unit P 2 +Q2 raises the polynomial degree by +2. This allows a quartic choice Φ = (Q2 + P 2)Q2, which may reduce to Φ = Q2, but also identifies with a valid perturbing term. The algebraic

2 2 1 2 Hamiltonian 2Hϕ(p, q) = (q + p )(1 q ) transforms canonically to its action-angle form, − 4 1 2 2 Hϕ(λ, φ) = λ λ sin(φ) , again quadratic in the variable λ. This geometry includes an − 2 oscillation disk in the energy range 0 < α < 1, which appears in the center of Fig. 2.8. By

design, the quarter period of Hϕ is K(α).

Isoperiodicity suggests that the two Hamiltonian models, Hϑ and Hϕ, will equate,

Hϑ(pi, qi) = Hϕ(pf , qf ), by a canonical transformation. A plausible guess that pi q˙i = pf q˙f 14It should be possible to prove this assertion by induction on n. 57 gives a second constraint equation in two sets of two unknowns. The non-linear system can be solved for either, 1 q 2 1 q 2 (pi, qi) = pf 4 q , arcsin qf 4 q or 2 − f 2 − f p p (pf , qf ) = csc(qi) pi 2 2 cos(qi), 2 2 cos(qi) , ± − −

and the Jacobian matrices are written out as,     2 2

p tan(q /2) √4−qf pf qf

p2 2 cos(q ) csc(q ) i i i i J    2 2 4−q2  J =  − √2−2 cos(qi)  and =  − √ f  , ±      sin(qi)   2  0 0 2 √2−2 cos(qi) √4−qf while requiring that qf √2. The domain restriction is permissive enough to map between | | ≤

entire oscillation disks. According to the determinants, det(J) = det( )J = 1, the angle φ is indeed a canonical coordinate of the pendulum Hamiltonian. Thus solution of the pendulum equations of motion follows from solution of the time parameterization problem on the phase

curves of either Hϑ or Hϕ. We will now pursue one particular solution in even more detail.

2.4. Incorporating Complex Time

Harold Edwards’s theory of elliptic curves and elliptic functions starts with an alternative normal form, 2G = (x2 + y2) a2x2y2 a2, which determines a family of elliptic curves, − − (a) = (x, y) : 2G(x, y, a) = 0 & a4 = 0 or 1 , all non-singular according to the extra C { 6 } condition on modulus a4. With a-dependence swept into τ, the crucial elliptic function,

2 P∞ eiπ(2n−1)2τ/2 cos (2n 1)πt ψ(t) = χ(t + 1 ) = n=1 − , 2 P∞ iπ(2n)2τ/2 1 + 2 n=1 e cos 2nπt allows a solution, x(t) = χ(t) and y(t) = ψ(t), of the coupled diﬀerential equations15, x˙ = y(1 a2x2) andy ˙ = x(1 a2y2), as derived from the (a) tangent geometry. Edwards − − − C already proves this statement in Part III of the original article [27]. Translation of his solution to the simple pendulum context encounters two diﬃculties. First, a transformation

15Edwards’s normalization of the time domain requires chain rule, ψ˙ = (2/T )dψ/dt, with real period T . 58 is needed to go from math coordinates (x, y) to physical coordinates (p, q)—but this follows quickly from the previous developments. Second, and with more eﬀort, calculation of the correct value for the period ratio τ depends upon ascertainment of the double-periodicity notion. To this end, the Wick rotation, t it, helps to extend transformation theory, and → to show relation between real and complex periods. In a sense, function G is Hamiltonian with canonical coordinates x and y. The analogy

2 2 4 can be improved. Rewriting (G, a , x, y) (H1, α, p, q) = (a G, a , a y, a x) produces → 2 2 2 2 a Hamiltonian form, α = 2H1(p, q) = p + q p q , with an exact parameterization − p(t) = α1/4 ψ(t), q(t) = α1/4 χ(t). The transformation does not leave the energy scale invariant from G to H, so it is not canonnical per se. Instead, the transformation

16 ∂αf ∂tf (αi, ti, pi, qi) (αf , tf , pf , qf ), is called covariant because , J = . In action angle → | | ∂αi ∂ti 1 2 2 coordinates, the Hamiltonian becomes H1(λ, φ) = λ λ sin(2φ) . Similarity between H1 − 2 and Hϕ suggests another covariant transformation, (ti, φi) (tf , φf ) = 2(ti, φi). Angle → doubling transfers the exact solution from H1 to Hϕ, where,

√α λ(t) = χ( t )2 + ψ( t )2 , φ(t) = arctan χ( t )/ψ( t ), 2 2 2 2 2

Time parameterization for pendulum variables then follows by walking back (λ, φ) through

the sequence of canonical transformations—ﬁrst to the Cartesian coordinates of Hϕ,

t 2 t 2 t t ψ( 2 ) χ( 2 ) 2 χ( 2 ) ψ( 2 ) p(t) = − 1 , q(t) = 1 , 2 2 t 2 t 2 t 2 t 2 1 + χ( 2 ) ψ( 2 ) 1 + χ( 2 ) ψ( 2 ) then to the Cartesian coordinates of Hϑ,

ψ( t )2 χ( t )2 2 χ( t ) ψ( t ) p(t) = 1 θ˙(t) = 2 2 , q(t) = 1 θ(t) = arcsin 2 2 , 2 t−2 t 2 2 t 2 t 2 1 + χ( 2 ) ψ( 2 ) 1 + χ( 2 ) ψ( 2 )

and ﬁnally to θ and θ˙, after multiplying by a factor of two. The expression for θ(t) reduces again according to a special case of the X addition rule, proven in Part II of [27]. The

0 t addition rules act linearly along the time dimension, so the choice of x = x = χ( 2 ) and

16The more liberal condition on J follows again from transforming Hamilton’s equations. | | 59 ψ ψ(0)

0 1 2 0 t

i i i Fig. 2.9.: Real-Valued Slices of the Elliptic Function ψ(t); I(t) = 0 and τ = i, 3 , 9 , 27 .

0 t y = y = ψ( 2 ) sets X = χ(t), and also, θ(t) = 2 arcsin α1/4 χt .

Yet this formal solution is worthless until we have deﬁned the period ratio τ as a function of

K(1−α) the energy parameter α. In context, the correct answer is that τ = i 2K(α) , but again, why? The real period,

I dφ T (α) = 4K(α) = , p1 α sin(2φ)2 − gives a first hint of double-periodicity, because it satisfies a second order ordinary differential equation (again, refer back to [59] Section 5). A proof of the assertion depends upon an

annihilator b and its certiﬁcate function Ξ, A

2 sin(2 n φ) b = 1 4(1 2α)∂α 4α(1 α)∂α and Ξn = . A − − − − 2 n 1 α sin(n φ)23/2 − The zero sum,

1 b ∂φΞn = 0, A ◦ p1 α sin(n φ)2 − − is trivial to verify by applying chain rule and reducing trigonometric terms. Integration H of any exact differential along a complete cycle yields a zero, including dφ ∂φΞn = 0, thus the claim, b T (α) = T (α) ∂α(4α(1 α)∂αT (α)) = 0, stands true. Transformation A ◦ − − 60 α 1 α leaves b invariant, so the general solution of the second order differential equation → − A can be written as T (α) = c1K(α) + c2K(1 α). This solution space includes two special − solutions, the real and complex periods, TR(α) = 4 K(α) and TI(α) = i2K(1 α), whose − ratio reproduces the suggested form τ = TI(α)/TR(α). The hint seems to have paid off, but leaves room for doubt and suspicion. We do not yet have a physical reason to complexify the time variable by extending from a one-dimensional real axis to a two-dimensional complex plane; however, mathematical function theory after Abel calls ahead of schedule for just such an abstraction. To account for additional complex degrees of freedom, we will build out a Riemann surface17,

2 4 (α) = (p, q) C : α = 2H(p, q) (v, w, x, y) R : α = 2H(x + iv, y + iw) , R { ∈ }'{ ∈ } around each phase curve (α). If transformation of canonical p and q variables introduces C determinant J C/ 0, 1 , then neither the two-form dp dq, nor the time scale dt, nor | | ∈ { } ∧ Hamilton’s equations will preserve. However, it is possible to cancel the extra factor J | | by scaling time covariantly, ti tf = J ti. Any transformation with unit modulus J → | | | | acts as rotation on the complex t-plane. In particular, the Wick rotation, ti tf = i ti, → goes by π/2 radians through the time plane, thus permutes real and complex axes. The simplest covariant coordinate transformation having J = i takes pi pf = i pi and rotates | | → (α) by π/2 radians through a four dimensional space C2 R4. An Abel-Wick rotation R ' is any covariant, linear transformation having J = i and ∂tf = i. Abel-Wick rotation is | | ∂ti a neat and very important trick, which ultimately allows calculation of the complex period

on doubly-periodic (α) by integration along a real-valued level curve e(α). Before moving R C ahead to non-trivial topology, it is worthwhile to work through an easy example. Real-valued theory distinguishes between circular points and hyperbolic points, while

complex-valued theory does not. Any transformation such as (pi, qi) (pf , qf ) = (i pi, qi) → changes constraints, from circular H◦ to hyperbolic He◦, and vice versa. It is possible to

17The nomenclature ”Kleinian Surface” may be better. Jeremy Gray claims that Felix Klein (1849-1925) was the ﬁrst to think of a complexiﬁed algebraic curve as ”a closed surface in its own right” [36]. 61 z

vw x y XY

x y Fig. 2.10.: A Few Depictions of a Genus Zero Riemann Surface. embed the entire harmonic hyperboloid into three dimensions, while losing only a phase degree of freedom via the map (v, w) z = √v2 + w2, → ±

2 ◦(α) = (p, q) C : α = 2H◦(p, q) R { ∈ } 3 2 2 2 ◦(α) = (x, y, z) R : α + z = x + y . −→ S { ∈ }

More explicitly, ◦(α) is obtained by considering rotational symmetry in the (p, q)-plane. S 2 2 Coordinates x = w = 0 and z = v may be chosen such that e◦(α) = (y, z): α = z + y C { − } 2 2 is a hyperbola connected to the base circle ◦(α) = (x, y): α = x + y at the turning C { } points (p, q) = (0, √α). Any other point (p, q) = (x0, y0) along ◦(α) is yet again a turning ± C point; however, in a rotated coordinate system. Point (x0, y0) connects circular ◦(α) to C a rotated copy of hyperbolic e◦(α), which falls in the hyperplane determined by (x0 u + C y0 v)(x0 y y0 x) = 0. − Two equivalent parameterizations, 2 ◦(α) = (p, q) = √α sin(t), cos(t) C : t C , R ∈ ∈ 2 and e◦(α) = (p, q) = √α sinh(u), cosh(u) C : u C , R ∈ ∈ of the harmonic hyperboloid follow from the solution of Hamilton’s equations, around a circular point and a hyperbolic point respectively. Wick rotation t u = i t equates → 62 I(t)

0 0 2π 4π R(t)

2 −

Fig. 2.11.: Genus Zero Harmonic Hyperboloid and Singly-Periodic Uniformization.

◦(α) and e◦(α) up to a π/2-radian rotation through the complex p-plane. While the R R usual trigonometric functions are periodic, their hyperbolic counterparts certainly are not. Taken together, periodicity along Rt and non-periodicity along Ru implies genus zero. This identiﬁcation introduces some cognitive dissonance, because the sphere is usually given as the standard form of a genus zero surface. In three dimensions, the hyperboloid, 3 2 ◦(α) = (x, y, z) = √α sin(t) cosh(u), cos(t) cosh(u), sinh(u) R :(t, u) R , S ∈ ∈

transforms into the Riemann sphere by stereographic projection,

2√α x 2√α y (α x2 y2) (x, y, z) X,Y,Z = , , − − , → α + x2 + y2 α + x2 + y2 ±(α + x2 + y2)

with + chosen for z 0 and chosen otherwise. ≥ − By having two periods rather than just one, a genus one elliptic curve diﬀers qualitatively from the genus zero harmonic hyperboloid. In general, the two periods may describe the boundaries of a period parallelogram in the complex t-plane; however, in Edwards’s normal form, a choice of real α (0, 1) amounts to choosing a period rectangle with entirely complex ∈ τ and I(τ) (0, ). To calculate τ explicitly, we transform by a Abel-Wick rotation, ∈ ∞ H1 He1, as in Fig. 2.12. Either transformation introduces a factor J = i relative to the → | | 63 2 2 2 2 2H1 = p + q p q − Abel-Wick Rotation ←− 2 2 2 2 2He1 = p + q + p q − 2 2 1 2 2 2 2He1 = p + q (p + q ) − − 4

Fig. 2.12.: Toric Cross Sections via Abel-Wick Rotation.

18 initial choice of coordinates . Constraint α = 2He1, for either He1, cuts out a real-valued curve e(α). Hamilton’s equations determine the complex period, up to a missing factor i. C One choice leads to an easier integral,

I Z ∞ dp TI(α) = i dt = i , p 2 2 −∞ (1 + p )(α + p )

on a non-compact contour, while the other oﬀers compactness with a harmonic limit ω = 2

as α 1. Integral value TI(1) = iπ also implies that ω = 2πi/TI(1) = 2. More importantly, → the zero sum,

1 p (1 + p2)1/2 b + ∂p Ξe = 0, with Ξe = , A ◦ p(1 + p2)(α + p2) (α + p2)3/2

R ∞ certifies that b TI(α) = 0, because dp ∂pΞe = 0. The harmonic limit determines the A ◦ −∞ constants of integration, and TI(α) = i2K(1 α), as desired. Depending on the reader’s − naiveté,it is either astounding or mundane that both real and complex solutions should satisfy the same differential equation. The cohomological theory of algebraic varieties gives a standard mathematical explanation as to why 19, but for now we are more concerned with the ever-pressing what for.

18Easy Exercise: Prove J = i explicitly, then prove the two alternatives canonically equivalent. 19Along any homology 1-cycle,| | integral H dt is rational, so satisﬁes a ”Picard-Fuchs type” ODE [64]. 64 t C 4τ ∈

2τ

0 0 2 4 Fig. 2.13.: Genus One Elliptic Curves and Doubly-Periodic Uniformization.

Unlike their trigonometric counterparts, the shape of χ(t) or ψ(t) depends on the amplitude

K(1−α) parameter α via the period ratio τ = i 2K(α) , now justly derived. Shape variation with α carries over to the Riemann surface,

n 1 2 o 1(α) = p, q = α 4 ψ(t), χ(t) C : t C , R ∈ ∈

which limits to a central harmonic hyperboloid when α 0 and I(t) 2n i τ, n Z. → → ∈ Around I(t) = 0, surface 1(α) looks suﬃciently like the harmonic hyperboloid to allow a R similar embedding20,

n 1 o p 2 2 3 τ 1(α) = x, y, z = α 4 ψR(t), χR(t), χI(t) + ψI(t) R : t R [0, ] . S ± ∈ ∈ × ± 2

Subscripts R and I indicate either the real or the imaginary component of complex-valued functions ψ(t) and χ(t). Hyperbolic sections also expand over thin-strip domains of width

∆t = 1/4, centered on vertical lines R(t) = (2n + 1)/4, n Z. Four sections in the range ∈ C2 separately approach a harmonic hyperboloid in the limit α 1. In the right panel of → Fig. 2.13, the partial domain of all five hyperbolic sections covers a full three-quarters of C while avoiding points (2m + 1)τ + n/2, (m, n) Z2. At these points, either χ(t) or ψ(t) has ∈ 20The map (p, q) (x, y, z), from to , preserves the euclidean metric: p?p + q?q = x2 + y2 + z2. → R S 65 a simple pole. Embedding the disjoint pieces of 1(α) into three dimensions and coloring R by the real component of time, we expand from the skeleton of Fig. 2.12 to the patchwork surfaces nested in the left of Fig. 2.13. Depiction of complex tori flexes impressive strength, but actually requires considerably more effort than is necessary for applications in classical physics. With only values of ψ(t) above the real time axis, we can graph sections of a Hamiltonian flow and make a wide range of experimental predictions—the two tasks are not entirely different. Fig. 2.14 depicts a section of the pendulum’s Hamiltonian flow,

1 1 ˙ 3 ϑ = θ(t), θ(t), 2K(α)t R : t R , F 2 2 ∈ ∈

along with the ﬂows ϕ and 1 associated with Hϕ and H1. In terms of textbook F F expectations, these graphs leave little to be desired, save a few numerical double-checks. The following zero sums,

0 = 2H(q(t), q(t)) α, 0 =p ˙(t) + ∂qH t, 0 =q ˙(t) ∂pH t, − | − | can be evaluated for real sample points t [0, 2] and α (0, 0.99). A truncation of ψ ∈ ∈ after five summation terms is sufficient to converge to machine precision, 0 10−15. This ≈ agreement should dispel any remaining doubt about veracity of the solution, but if necessary, testing can be extended to higher values α (.99, 1) and to arbitrary precision simply by ∈ including more summation terms. Even then, those more inclined to standard methodology can be expected to issue an obstinate argument that ”Hamilton’s equations have a unique solution. For the simple pendulum, that solution is written in terms of the Jacobian Elliptic functions”. Certainly this is also true [99, 100], and by comparison of equally valid solutions, we find that,

t 2 t 2 − 1 ψ( 2 ) χ( 2 ) sn 2K(α)t α = α 4 χ(t), cn 2K(α)t α = , t 2 − t 2 χ( 2 ) + ψ( 2 ) only where α (0, 1) and τ = i K(1−α) (0, ). These identities imply that any equations ∈ 2K(α) ∈ ∞ solvable in terms of the Jacobian elliptic functions are also solvable in terms of ψ(t) with 66 1 ϕ ϑ F F F

p p p 1 1 1

t t t

q π q q 1 √2 2

t t t

Online: Edwards’s Solution of Pendulum Oscillation

Fig. 2.14.: Hamiltonian Flows and their vertical Projections over one Period.

67 Fig. 2.15.: A Modiﬁed Fidget Spinner.

entirely imaginary τ. More work remains to be done. Wherever a standard solutions exists and is known, another alternative solution in terms of ψ(t) is waiting to be found. What we have seen so far encourages the hope that new derivations will lead to even more new insight!

2.5. Experiment and Data Analysis

A fidget spinner consists of a low friction ball bearing, a pair of finger grips, and a weighted rigid body. The ball bearing connects the finger grips on its interior to the rigid body on its exterior. Typically the rigid body has three symmetrically spaced lobes, each with a pendulous mass exterior. Added masses are balanced to maximally conserve angular momentum when the spinner is in motion, regardless of orientation relative to the ambient gravitational field. Recently, fidget spinners became a trend in the novelty market. They are now available in surplus for classroom demonstrations or laboratory experiments. To adapt a fidget spinner to our needs, we remove the finger grips and two of three exterior weights, as in Fig. 2.15. Attaching the bearing interior to a fixed horizontal axle completes construction of a compound pendulum. For each trial, we deflect the pendulum to its maximum energy configuration and observe

68 Fig. 2.16.: Reduction of a data frame to Cartesian position coordinates.

libration decay to zero total energy. Quality of each trial depends on energy loss through vibrational coupling to the surrounding environment. Heuristically, longer running trials are expected to yield higher quality data. We utilize a fixed and immobile camera21 to record each trial at 240 fps. Tens of thousands of data points are recorded while the fidget spinner is in motion (for only 2 - 2.5 minutes). Each data point starts as a frame of the video, or equivalently, as a digital RGB image. The set of all frames can be retrieved from the video file using a command line utility such as ffmpeg. After ripping the frames, we obtain position vs. time data by image analysis. Prior to the experiment, we decorate and illuminate the spinner so that transforming an image to grayscale and thresholding to black and white leaves only a motion-tracking spot. The pixel average across the spot determines a single Cartesian position for the remaining pendulous mass, as in Fig. 2.16. Once all frames are changed into position data, points from the dataset fall onto the arc of a circle. Then corresponding angular displacements are obtained using the arctan function. One could try to fit angle vs. time data using elliptic functions, but such analysis is usually overkill. Instead, we redact the angle vs. time data to period vs. energy data by associating successive local maxima in kinetic energy with the

21Many cell phones now have slow-motion video capture, for the present experiment we used an iPhone 6. One trial of data is available through Youtube: https://www.youtube.com/watch?v=DMDdykItGlo. 69 α 0.5 2T0 T π θ ≈ K(α) 2

5s 3 2 T0 t

π α 2 T − 0 10 Fig. 2.17.: Data Redaction. Five seconds of θ(t) (left), Period vs. Energy (right).

time intervals between them22. Each pair of successive maxima have a small diﬀerence, say

∆E = E1 E0. So long as ∆E remains small relative to the mean of E0 and E1, we safely − assume that the data point describes an energy-conserving motion. More rigorous quality analysis is advisable. As a measure of left/right asymmetry, we have compared successive ∆E in detail, but omit explanation here for sake of brevity. As common sense would dictate, losses dominate at low and high energy. For the dataset of Fig. 2.17, most acceptable ﬁdelity restricts to the energy range α [0.05, 0.6] [0, 1]. ∈ ⊂ Before analyzing the data, ﬁrst we must realize that the Hamiltonian is not exactly Hϑ = p2 + sin(q)2. The rigid body connecting the bearing to the pendulous mass has a moment

1 ˙2 of inertia, call it Iz¯. The kinetic energy needs an extra term like 2 Iz¯θ so that KE = 1 2 ˙2 2 (Iz¯ + ml )θ . However, the extra rotational term contributes next to nothing because the mass of the rigid body is relatively small. Let m be the mass of the pendulous mass, and

2 l its hanging distance. The inequality ml Iz¯ says that kinetic energy is dominated by the hanging mass, so the system can be treated approximately as a simple pendulum. The easiest and most obvious data analysis ﬁts the complete elliptic integral of the ﬁrst kind, K(α), to period vs. energy data. For the data set of Fig. 2.17, the period is extracted as p 104 frames, which is close to the estimate of 108 frames, from T0 2π l/g with l 5 cm. ≈ ≈

22Where angle vs. time data reaches maximum velocity, ﬁt a linear function, and take the slope. Slope squared is proportional to kinetic energy. 70 The harmonic frequency T0 locates an intercept on the vertical axis, but says nothing about the shape of the period vs. energy data. In non-linear analysis, we mostly disregard the harmonic period and instead seek to characterize the functional shape. A shape parameter controls the curvature of a ﬁt function, and must be extracted from data to test agreement

with theory. Edwards’s model H1 is notable for its square dihedral symmetry, but it is not the only square-symmetric model of quartic order. Adding in a perturbation like (p2 + q2)2, we obtain an entire stratum, 2 2 1 2 2 2 2 2 Σ = (p, q): α = 2H() = p + q (p + q ) + p q ; α [0, 1); [1/2, 2] , − 4 ∈ ∈ as depicted in Fig. 2.18. In action-angle coordinates, the Hamiltonian energy function is written as 2H() = 2λ λ2 + λ21 cos(4φ). The double-cover from 4φ 2φ transforms − 2 − → H(α, ) to a less symmetric form, 1 1 2H0(α, ) = p2 + q2 (p2 + q2)2 + (p2 + q2)p2, − 4 4 but preserves period function T (α, ) up to a scale factor of 2. Hamiltonian H0() includes

2 2 1 2 2Hϕ = (p + q )(1 q ) at = 1. Changing the shape parameter does not aﬀect the − 4 energy domain, but also changes the shape of the oscillation disk and its corresponding period function. By ﬁtting T (α, ) to data, we quantify how well the experimental oscillation follows

0 the theoretical expectation that H () = Hϕ, i.e. that = 1. After root-solving for λ, period function T (α, ) can be written in integral form, I dφ T (α, ) = q . 1 α1 sin(φ)2 − − To solve T (α, ) in closed form, we make use of another annihilator/certiﬁcate pair23,

2 2 b() = 2 3α + 3α 4(1 4α + 3α + 2α 3α )∂α A − − − − − 4α(1 α)(1 α + α)∂2 , − − − α cos(φ) sin(φ) Ξ = − , 3/2 1 α1 sin(φ)2 − − 23Given valid input, either algorithm ExpToODE or DihedralToODE will output the annihilating operator. These algorithms will be described in a subsequent chapter. 71

3 2 3 α = 4 1

1 2

Fig. 2.18.: The stratum Σ.

dT such that b() ∂φΞ = 0, and the exact diﬀerential ∂φΞ vanishes in the integral over A ◦ dφ − one complete cycle, so b() T (α, ) = 0. When presenting b() one other remark should A ◦ A always be made. Simultaneously aﬀecting α 1 α and 1/ leaves b() invariant, → − → A incredible! The two, linearly-independent solutions of b() can be written as, A 2 α 2 α 1 TR(α, ) = K and TI(α, ) = K − , π√1 α α 1 π√α α − −

so that the invariant transformation permutes TR TI. Both solutions are normalized to ↔ have TR(0, ) = TI(1, ) = 1, so the scaled real period is written as T (α, ) = T0TR(α, ). After the data is recorded and the functional model is identiﬁed, the process of parameter extraction begins by deﬁning a likelihood function, 2 X T0TR(αi, ) ti (T0, ) = exp − , L − σ2 i∈data i

with (αi, ti) period vs. energy data, and σi a standard error. Perhaps with too much expediency, we simply set each σi to statistical standard deviation, after binning data with ﬁve consecutive points in each bin. This reduces the total data count from 500 to 100. ∼ ∼ After calculating values of (T0, ) on a square grid, 1σ, 2σ, and 3σ contours are calculated L 72 T 2T0 17.5% ) 5% ∼ . 0

3 Time Scale ( 2T0

1 0.98

α T0 1 0 2 1 Fig. 2.19.: Extraction of shape parameter = 0.98 and the best ﬁt curve T (α, ). ≈

and plotted in the inset of Fig. 2.19. The likelihood function is single-peaked and slightly

covariant between T0 and . Projection onto the axis produces an estimate = 0.98 0.03, ± which is correct within 1σ. We have taken error analysis too lightly, erring toward an overly conservative error estimate. It can be argued that the uncertainty estimate is too large, because the dashed lines of Fig. 2.19, plotted at 1σ, look to include all data points. Under normal assumptions, a handful of data points could be expected to fall between the 1σ and 3σ bounds. Even with a tighter error analysis it seems entirely improbable that the solid outer curves at 3σ would exclude K(α) at = 1. We take the present result as satisfactory, and reiterate our role as pathﬁnders. It is not really to collect the best data or to worry too much about scrutinizing the data; instead, we simply aim to originate a new direction for non-linear analysts to follow. If anyone desires to try and tighten the error bound, good for them, and they can probably improve upon the result of Fig. 2.19. Our prerogative moving forward is to shed more light on the theoretical question, ”Where do the Annihilators come from?”

73 2.6. Conclusion

In mathematics and physics, choice of coordinates always matters. We have demonstrated this adage yet again by exactly solving the equations of motion of the simple pendulum. Coordinate transformation enables rapid analysis of period integrals in terms of an ordinary differential equation paired with one or more coordinate-dependent certificate function(s). Our focused discussion of the simple pendulum only pertains to one oscillation disk and its variant forms. The harmonic hyperboloid and the pendulum’s anharmonic tori are only the first examples on a magnificent bridge between physics and mathematics. The dream of ”Hamilton-Abel” theory is to see more pedestrian and commercial traffic crossing this bridge. An obvious next step is to analyze various similar oscillation disks in terms of their period functions. The prospectus of the dissertation prelude [59] gives many more related examples. These examples are specially chosen to enable more useful work in physics, including analysis of experimental data. A wide variety of oscillation disks will succumb to analysis by creative telescoping, so it is well-worth learning about annihilators and certificates at the start. Again adequate choice of coordinates plays an important role in our subsequent development of algorithmic means. Using symbolic computation, it is possible to automatically identify an oscillation disk by the ordinary differential equation its period function must satisfy. Subsequent questions of isoperiodicity can then be answered affirmative or negative. In fact, it is desirable that alternative candidate models are found to differ. In an oscillation experiment, period-energy data then can suffice to decide between candidates, as we have seen in the final section. If classical mechanics is the only domain of concern, complex period functions do not help to explain real-valued measurements, and the technique of Abel-Wick rotation does not attain its full importance. Instead of continuous period functions, quantum mechanics deals with discrete, quantum values. In the type of semiclassical calculations that owe back to the Wiemar time of Ehrenfest and Sommerfeld, quantum values are grossly determined by root solving a real action function, but such a procedure does not explain every detail. So-called

74 splitting energies also need to be calculated using complex transformation theory. When we ﬁnally get to complex-time tunneling integrals along Riemannian tori, Abel’s original insight on double-periodicity will seem all the more profound and prescient.

Acknowledgements

Sometime before this paper was written, Andrew N.W. Hone made a helpful suggestion about Hamilton’s equations via [seqfan] mailing list. Calculations and drawings are the author’s own work (usually with the aid of Mathematica [54]).

75 3. Geometric Interpretation of a few Hypergeometric Series

Ramanujan’s article ”Modular equations and approximations to π” draws special attention to three curious hypergeometric series. In mathematical physics, each series determines periods along the elliptic level curves of an integrable Hamiltonian surface. We perform a combinatorial search of Hamiltonian function space and find a diverse collection of closely related species, some of which may have been unknown previously. The search is made rigorous by here defined diagnostic algorithms ExpToODE, HyperelipticToODE and DihedralToODE. All three take an input integral function and output the relevant ordinary differential equation. Examples growing out of Ramanujan’s theory have intrinsic value, but they are not the only interesting use cases of the diagnostic algorithms. What we can learn from this ”walk through Ramanujan’s garden” will ultimately help us to extend systematic analysis elsewhere.

3.1. History and Introduction

The aphorism ”history tends to favor the victors” applies no less in particular to history of science. Ideally, science should be an apolitical process of ﬁnding out the truth by reason and experimentation. Practically, we can always expect to hear another anecdote of unfair practice and needless exclusion, even if we only listen to the inner workings of the dominant western system. Eurocentrism and Anglocentrism present an even more signiﬁcant problem to the world. Western scientists can, and often do, ignore developments and contributions of non-western origin. Our discussion of classical science thus far repeats the mistake of exclusive biasing by focusing solely on European lineages [58, 59]. We will now begin to correct for this mistake by discussing a few earlier developments of India. The biased1 slogan Asian ideas matter is especially relevant when we grapple with even a small part of the theory of Srinivasa Ramanujan (1887-1920). From what we know about

1Compare with ”Hsin Shin Ming”, ”Nagarjuna’s four propositions and zen”, available at https://kansaszencenter.org/resources/. 76 Ramanujan’s heritage, we can easily understand that his life and work built upon thousands of years of Indian culture2. To buck Anglocentrism, we can look at Ramanujan’s notebooks, and then ask, which of his basic ideas and techniques were already known in India before the invasion of the British East India Trading Company? Less than two decades after Ramanujan’s untimely death, A.N. Singh published an imminently useful, English-language review, ”On the use of series in Hindu mathematics”3[88]. This article clearly shows that Ramanujan was not the first Indian scientist to take an interest in binomial coefficients, series expansions, or the transcendental constant π. That being said, Ramanujan was among the first to work directly with the western scientific world. Two histories need to be compared.

n The binomial coefficients are an irrevocably important set of numbers denoted k , usually for integer n and k with 0 k n. In combinatorics, n determines the number of ways to ≤ ≤ k n n! choose k distinct elements from a set of cardinality n, and the identity k = (n−k)!k! counts out the answer in terms of the factorial function n! = n (n 1) (n 2) ... 1 and 0! = 1. × − × − × × n n n n−1 n−1 Algebra gives an alternative, additive definition, 0 = n = 1 otherwise k = k−1 + k , n Pn n k Pn−1 n−1 k which follows from the expansion (1 + z) = k=0 k z = (1 + z) k=0 k z . It is quite natural to view the binomial coefficients as elements of a triangular array, as in Fig. 3.1. When the integers are arranged in this particular way, each row determines each next row by the addition rule. Such an arrangement often goes by the name ”Pascal’s triangle”. A.N. Singh tells us that the same tabulation was derived in India under a poetic heading, Meru Prasta¯ra, ”the Steps of Meru”, possibly as early as Pingala’s time circa 200 BC. In Indian cosmology, Meru plays a role similar to that of Parnassus in Greek cosmology—It is the center of the poetic universe4. Pingala asked the question, given a verse of n syllables divided into k lagu (light), and n k guru (heavy), how many orderings are possible? Not only did Pingala − Pn n n know the answer in terms of Meru Prasta¯ra, he also knew the row sums k=0 k = 2 [88].

2Robert Kanigel has written a biography of Ramanujan, The man who knew inﬁnity. The book was recently adapted to a motion picture under the same title. 3A scan is available through Erv Wilson’s electronic library, http://www.anaphoria.com/library.html. 4Compare Aryabhatiya Section 4 Items 11-12 with ”Sumeru” of the Vimalakirti Nirdesa Sutra. For the West-East connection, one can start by ﬁnding out about the Omphalos of Delphi. 77 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Fig. 3.1.: Pingala’s Meru Prasta¯ra.

This was an important discovery for Indian science, and thereafter Sanskrit prosody became a standard format for Vedic science. Over hundreds and thousands of years, Vedic science mobilized throughout the entire Indian subcontinent, most lately developing in the southern states of Kerala and Tamil Nadu5. The Kerala school of Astronomy and Mathematics is associated with the life and work Madhava of Sangamagrama (circa 1350-1425)6. It is now commonly accepted that Madhava originated the poetic definition of series expansions for sine, cosine[79]. At the time, the standard educational procedure was to transmit knowledge mainly by spoken word. The teacher would recite his verses directly to the students, and rigor would only be done if the listeners could think of an objection to the master’s teaching (Watch [72], and S.P. Singh concurred by email). Due to this practice, the written record is not as helpful as we would like; although, over time, more details became available through the notes of subsequent students working in the lineages at Kerala. Their accomplishment is too profound to belittle by imposing an exogenous western bias. Without any notion of calculus, Indian mathematicians at Kerala preceeded the greatest western scientists at the central task of exactly defining the transcendental constant π in terms of an infinite series expansion[79]. Given that knowledge was already sufficiently mobile and moving southward, it is entirely

5For an overview see George Gheverghese Joseph’s The Crest of the Peacock, Chapters 8-10. 6A few short biographies are available through MacTutor: Aryabhata, Madhava, Ramanujan. 78 probable that Vedic science had taken hold in Tamil Nadu by XIX Century. Ramanujan was born in Tamil Nadu on December 22, 1887, and attended primary school there. Were benevolent Hindu nationalists able to influence Ramanujan’s early development as a mathematician? We can only say that Ramanujan, as a member of the Brahmin caste, had a right to receive Vedic science, not that he actually did. At this time, the traditions of India became undermined by British interventionism, which included a commandeering of the educational system. During the same years, Mahatma Gandhi (1869-1948) developed the strategy of Satyagraha7, which would eventually win back Indian independence, though not during Ramanujan’s short lifetime. Sadly, we do not know if Ramanujan was aware of Pingala or Madhava by the start of his research into western mathematics. Regardless of what resources he could access in the material world, Ramanujan became the natural successor of India’s earlier number theorists. Instead of linguistic poetry, he wrote poetic equations. His Indian notebooks are filled with thousands of cryptic entries, which have resisted attempts to translate or decipher. Regrettably, the effort to understand Ramanujan’s educational circumstances started only after his death, with the obituary of Seshu Aiyer and Ramachandra Rao, and Hardy’s short biography ”The Indian mathematician Ramanujan”[43, 81]. Neither article says anything about Pingala or Madhava, but both mention Euler in connection with Ramanujan’s early interest in sine and cosine functions. Hardy’s account also claims that ”Carr’s Synopsis. . . first aroused Ramanujan’s full powers”. We don’t know if this Hardy assertion is true, but it sounds possibly overstated due to Anglocentric bias. Nevertheless, it is an important historical fact that Ramanujan studied Carr, because Synopsis defines figurate numbers next to hypergeometric function 2F1 in items 289-292. In fact, Item 290 gives a tabulation, in a slightly different form, of the first few rows of Pascal’s Triangle. Carr, an Englishman, mentions neither Pascal nor Pingala. Even if Ramanujan was unaware of

7In his monumental ”I Have a Dream” speech, Martin Luther King Jr. echoed Ghandi by saying ”Again and again we must rise to the majestic heights of meeting physical force with soul force.” Audio recording and transcript available online at https://kinginstitute.stanford.edu/king-papers/. 79 his own nation’s historical contributions to science, he would have found the same ideas in Carr; admittedly, with wrong attribution. For a period of time, interest in Ramanujan languished. In the late 1970’s, Bruce Berndt and coworkers revived the mission of editing Ramanujan’s notebooks. An overview of their work is given in [13]. It shows that Ramanujan was interested in many topics, and that he paid quite a lot of attention to the theory of elliptic functions. According to a bibliography of Ramanujan’s primary sources [10], he had access to A.G. Greenhill’s treatise on elliptic functions8, and at first, found some interest in the period function of the simple pendulum, the complete elliptic integral of the first kind9, here denoted by symbol K(α). Legendre found an identity, K sin(5π/12)2 = √3K sin(π/12)2, which piqued Ramanujan’s interest and led him to record a few original results in his notebooks [97] (and see also item 13 of Hardy’s obituary [43]). Not only did Ramanujan at times exceed the preexisting theory of elliptic integrals, he also broke completely free of its confines. Section 11 of Berndt’s overview pertains to Ramanujan’s theory of alternative bases, which is also described in two full length articles [7, 11]. An important takeaway from these accounts is that entries of Ramanujan’s notebooks show he had already discovered existence of the three alternative theories while living in India. This does and does not help to explain how he could rapidly publish the article ”Modular Equations and approximations to π”, much less than one year after arriving at Cambridge University10[77]. Following the poetry of Freeman Dyson (1923-2020), Berndt also wrote a lovely invitation to catalogue the ”Flowers which we cannot yet see growing in Ramanujan’s garden of hypergeometric series, elliptic functions, and q’s”11 [12]. In this work of outlook (or is it actually introspection?), mathematical loose ends of Ramanujan’s theory of alternative bases take eminent position in section 2. There it says,

8Scans are available online via archive.org, including Carr’s Synopsis and Greenhill’s Elliptic Functions. 9We have already rigorously deﬁned K(α) in [58, 59]. Another certiﬁcate is given in Section 4. 10 A complete bibliography is available online at https://www.imsc.res.in/∼rao/ramanujan/. 11Our own thoughts are along the lines of ”Parametric curves / which cannot yet be seen to grow / in a garden of Mystery”, but we will defer to historical precedent until a better consensus can be reached. 80 In his famous paper [Modular equations and approximations to π], Ramanujan records several elegant series for 1/π and asserts ”There are corresponding theories in which q is replace by one or other of the functions” 1 r−1 2F1( r , r ; 1; 1 x) qr := qr(x) := exp π csc(π/r) 1 r−1 − , − 2F1( r , r ; 1; x)

where r = 3, 4, or 6 and where 2F1 denotes the classical Gaussian hypergeometric function.

Neither in the original notebooks, nor in the journal article, nor in subsequent editing did explicit mention of elliptic curves play a very signiﬁcant role. However, Berndt and colleagues took a signiﬁcant step in this direction by changing Ramanujan’s Euler-inspired notation,

1 3 2 1 3 5 7 4 1 3 5 7 9 11 6 K1 = 1 + · k + · · · k + · · · · · k + ..., 42 42 82 42 82 122 · · · 1 2 2 1 2 4 5 4 1 2 4 5 7 8 6 K2 = 1 + · k + · · · k + · · · · · k + ..., 32 32 62 32 62 92 · · · 1 5 2 1 5 7 11 4 1 5 7 11 13 17 6 K3 = 1 + · k + · · · k + · · · · · k + ...,, 62 62 122 62 122 182 · · · to that of the standard hypergeometric theory. Since we know that the hypergeometric function, whether it is due to Gauss or Euler, is the solution of a second-order ordinary diﬀerential equation(Cf. [61], Section 2), we can immediately hypothesize that Ramanujan’s original assertion is equivalent to an assertion that

There are corresponding theories in which the underlying elliptic curve geometry

is replaced by one or other of the curve families 3(α), 4(α), 6(α). X X X

In this hypothesis the unknowns s(α) are Riemann tori over the complex numbers. As their X shape varies with α, the real and complex periods must be solutions of the hypergeometric diﬀerential equation12,

2 2 2 s Ts(α) = 0, where s = (s 1) s (1 2α)∂α s α(1 α)∂ . A ◦ A − − − − − α For s = 2, 4 and 6 geometric models were known classically, and maybe a model for the diﬃcult case s = 3 was also known to a small cadre of European cognescenti(?). Nevertheless,

12Our notation is different: r is reserved for radius, s is for signature, and α is the default expansion parameter. For a detailed working of case s = 2, see [59] Section 5 and [58]. 81 a systematic exposition was missing from the literature, until the issue was taken up by L.C. Shen. In a series of articles [82, 83, 84, 85] he revealed that the relevant geometries could be obtained from the Chebyshev polynomials, amazing! We may not be too surprised to find that the mysterious garden contains even more as yet unseen. The unmentioned alternatives have different symmetries, different genera, and perhaps even different fragrances, so they are also deserving of correct and individual diagnosis. Presently the theory of Creative Telescoping gives us new tools for systematizing and automating analysis of integrals taken along deformable curves13. In a most general form, as recently discussed by Bostan and Lairez, the complications are many [21, 64]. However, when it is possible to reduce a geometric integral to a univariate form, simple Hermite reduction can be used effectively. The goal is to take a curve, say (α) = (p, q) : 2H(p, q) = α , X { } H write the period integral T (α) = X dt, and instead of evaluating T (α) directly, to find

an annihilating operator Q[[α, ∂α]] (also called a telescoper) such that T (α) = 0. A ∈ A ◦ Annihilation of the period function happens when dt ∂tΞ = 0, with certificate Ξ a A ◦ − function of t. If the annihilator and its certificate are known, the task of evaluating the integral T (α) can be changed for a similar task of solving an ordinary differential equation. This is a much easier approach and allows us to efficiently search for special (α). X In two previous articles (or chapters), we have already discussed the standard case s = 2

h 1 1 i 2 , 2 in detail[58, 59]. The appropriate period function, T2(α) = 4K(α) = 2π 2F1 1 α , can P 1 2n2 n be written in terms of the central binomial coefficients T2(α) = 2π n≥0 16n n α . If we 1 throw away the denominators 16n , and search the Online Encyclopedia of Integer Sequences (OEIS)[89] for the integer expansion coefficients, then we find entry A002894. Clicking through the cross references, we might also find A006480 and A000897, and both entries cross reference to A113424.Though historically unusual14, this is still a great and easy way

13For a list of relevant references, trackback from the list given in [59] and see also [19]. 14When, before now, have we ever made such interesting maths discoveries by clicking a hyperlink? 82 to ﬁnd out more about Ramanujan’s alternative series,

1 2 3 , 3 X 1 3n 2n n T3(α) = 2F1 α = α , 1 27n n n n≥0 1 3 4 , 4 X 1 2n 4n n T4(α) = 2F1 α = α , 1 64n n 2n n≥0 1 5 6 , 6 X 1 3n 6n n and T6(α) = 2F1 α = α , 1 432n n 3n n≥0

in Ramanujan’s notation K2, K1, and K3 respectively. Written in this way, the dependence on Pingala is clear, but mystery remains. Why should these particular binomial products matter at all, and why only these three? These questions are motivation enough for the present work, but a little more will be said about π and the Kerala connection. When calculating and storing proof data on a computer, we are not limited by paper shortages. Nothing but time will prevent us from recording as many results as we can. The combinatorics of Section 2 can be skipped, but it leads quite naturally into Section 3 where most of the rigorous analysis is done by a few different implementations of an algorithm EasyCT. Requiring concordance between alternative implementations, not only do we easily find three relevant families of elliptic curves, we find that these three are somehow the lone minimal examples (except for a few higher-dimensional reflections). New geometric definitions allow quick and easy proof of Legendre-style identities, which gets us into Chapter 1 of ”π and the AGM”[18]. It is tempting to keep going in a pure maths direction, but by the end, we will need to steer ourselves back toward physics calculations. Again, mathematical loose ends may be appreciated for what they are worth.

3.2. Creative Combinatorics

We have already shown that a few diﬀerent Hamiltonian functions determine the simple pendulum’s libration behavior, and that transformation theory accounts for equivalence between the alternative forms [58]. Now it is due time to continue developing Hamiltonian mechanics by considering oscillation disks in more generality. As a breif reminder, an 83 2 3 4 6 C C C C n o s(α) = (p, q): α = 2Hs(p, q), s = 2, 3, 4, or 6 with α (0, 1). C ∈

h 1 s−1 i s , s Fig. 3.2.: Elliptic oscillation disks, with Ts(α) = 2π 2F1 1 α for s = 2, 3, 4, 6.

oscillation disk is a topological disk taken from the phase plane, which is bounded at its center by a circular point, and bounded on its outer edge(s) by a separatrix curve and at least one hyperbolic point15. Figure 3.2 shows four examples with blue level curves. The four

geometries diﬀer in symmetry; however, any curve s(α) is an elliptic curve of genus one. C In fact, the curves 2(α) to far left are determined by the physicist’s version of Edwards’s C 2 2 2 2 normal form[27], 2H2 = p + q p q . For other examples, the assertion of genus 1 can be − proven a few diﬀerent ways using the standard theory of elliptic curves16. Let us start with the more familiar cases where kinetic energy and potential energy are

1 2 separable, i.e. cases that have H(p, q) = 2 p + V (q) in units where m = 1. Choosing the 1 2 1 PN n 2 2 potential energy function V (q) = q + vnq forces a circular limit, 2H(p, q) p +q , 2 2 n=3 ≈ around the origin. The harmonic period at the circular point is T0 = 2π, so the angular

frequency scale is ω0 = 1 for the entire oscillation disk. Two exceedingly simple examples of this form are the cubic and quartic anharmonic oscillators,

2 2 2√3 3 2 2 1 4 2H6(p, q) = p + q q , 2H4(p, q) = p + q q . − 9 − 4 √ 2 3 1 Coeﬃcients v3 = and v4 = are chosen so that 2V (√3) = 1 for the cubic function, − 9 − 4 and 2V ( √2) = 1 for the quartic. Values q = √3 and q = √2 determine the only local ± ± 15For more deﬁnitions and theory, refer back to [58] section 3. 16All non-singular cubic plane curves are elliptic due to the existence of chord-and-tangent addition rule [86, 87]. The other quartic H4 is birationally equivalent to a cubic function, read on for more details. 84 maxima of either potential. These conventions must have a separatrix curve at α = 1 and

an oscillation disk with domain α [0, 1). To distinguish that H6 and H4 are not equivalent ∈ by canonical transformation, we need only calculate distinct period functions. In action-angle coordinates the Hamiltonian function of the quartic anharmonic osicllator

4 2 4 1−√1−α sin(φ) is written as 2H4(λ, φ) = 2λ λ sin(φ) . Solving α = 2H4(λ, φ) for λ = 4 − sin(φ) H H dφ then allows us to write the period integral, T4(α) = 2(∂αλ)dφ = . We should √1−α sin(φ)4 already know how to solve this integral in series expansion, I ∞ I ∞ dφ X 1 2n 4n X 2π 2n 4n T4(α) = = sin(φ) dφ = . p1 α sin(φ)4 4n n 64n n 2n − n=0 n=0 As with the previous cases E(α) and K(α), the expansion coeﬃcients, call them fn, satisfy a hypergeometric recursion, 2 1 3 1 2n 4n f0 = 1, (n + 1) fn+1 = (n + )(n + )fn fn = , 4 4 ⇐⇒ 64n n 2n

h 1 3 i 4 , 4 which determines a standard expression, T4(α) = 2π 2F1 1 α . It is instructive to compare

T4(α) with the earlier T2(α) = 4K(α) by writing the Hadamard products, 1 1 1 1 1 3 1 1 3 2 , 2 2 , 2 , 4 , 4 2 , 4 , 4 2F1 α = 2F1 · α ? 2F1 · α ; 2F1 α = 2F1 · α ? 2F1 α , 1 1 1 · · · 2 where the ? operator indicates multiplication of expansion coefficients17, ∞ ∞ ∞ X n X n X n F (α) ?G(α) = fn α ? gn α = fn gn α . n=0 n=0 n=0 The shared form, 2H = 2λ λ2Φ, determines the equivalent first factor, and they differ on − the second factor P αn H Φndφ. This observation suggests a first combinatorial foray. The condition of separable potential and kinetic energy requires Φ sin(φ)4. If we loosen ∝ this condition, then Φ need only be a trigonometric polynomial of homogeneous degree 4,

P4 n 4−n i.e. must have the form Φ = n=0 cn cos(φ) sin(φ) . By searching the range of valid perturbations Φ, it is possible to ﬁnd at least one more well-related case, 1 2H0 = p2 + q2 (p2 + q2)2 + 2p2q2 Φ = cos(φ)4 6 cos(φ)2 sin(φ)2 + sin(φ)4. 4 − 4 ⇐⇒ − 17When F and G are both hypergeometric functions: join upper parameters, join lower parameters with an additional 1, and ﬁnally pairwise cancel parameters occurring in both upper and lower sets. 85 0 0 0 C3 C4 C6

h 1 s−1 i s , s 2 Fig. 3.3.: Higher genus oscillation disks with T (α) = 2π 2F1 1 α for s = 3, 4, 6.

This case stands out because perturbing term reduces to Φ = cos(4φ). The oscillation disk takes the shape of a square with concave edges, as in the center of Fig. 3.3. On the disk,

h 1 3 i 2 4 , 4 2 time is measured by period function T4(α ) = 2π 2F1 1 α . To see why this coincidence should occur, observe that odd powers of Φ integrate to zero, while H cos(4φ)2ndφ 2n. ∝ n Relative to the case Φ = sin(φ)4, left and right Hadamard factors of the period function transpose—an impressive dance of parameters and exponents! We will encounter similar symmetry and similar maneuvers as we continue to study simple period functions. The cubic oscillator is terribly more diﬃcult to solve. In action-angle coordinates, the √ 4 6 3/2 3 3/2 deﬁnition 2H6 = 2λ λ cos(φ) is a special case of 2H = 2λ λ Φ. Constraint − 9 − α = 2H implies generally that αΦ2 = 2x x3/2, x = λΦ2. Exact root-solving is too unwieldy − a process, so we have no better recourse than series reversion (cf. A214377). Here we will drop rigor, skip the expansion of λ, and instead proceed directly to assert that,

∞ 2n ∞ X 1 3n I 4√6 X 1 3n6n T (α) = cos(φ)3 dφ = 2π . 6 8n n 9 432n n 3n n=0 n=0 A relatively easy proof will be given in the next section. For now, let us note the similarities

h 1 5 i 6 , 6 to previous examples. The standard expression T6(α) = 2π 2F1 1 α follows from, 2 1 5 1 3n 6n f0 = 1, (n + 1) fn+1 = (n + )(n + )fn fn = . 6 6 ⇐⇒ 432n n 3n

h 1 , 5 i h 1 , 2 i h 1 , 1 , 5 i 6 6 3 3 6 2 6 As before, Hadamard decomposition 2F1 1 α = 2F1 1 α ? 3F2 1 2 α combines a 2 3 , 3 86 left factor determined from the general cubic constraint, αΦ2 = 2λ λ3/2, with a right − factor determined by integrating powers of Φ. Again we can search valid choices of Φ to ﬁnd √ 2 2 4 1 3 2 4 6 another interesting case, 2H3 = p +q ( ) 2 (q 3p q), with Φ = sin(3φ). The period − 27 − 9 function can be calculated simply by changing the right factor of the Hadamard product.

h 1 2 i h 1 2 i h 1 , 2 i h 1 i 3 , 3 3 , 3 3 3 2 ,· That is, T3(α) = 2π 2F1 1 α follows from 2F1 1 α = 2F1 1 α ? 2F1 · α . 2 By the most expedient, intuitive analysis, we have already uncovered a secret that is well within the reaches of what Ramanujan himself could have known and calculated:

The three alternatives to K(α) all obey a Hadamard decomposition to two hypergeometric factors. One factor is determined by the general choice of a degree, either quartic or cubic. The other factor is then determined by the special choice of a trigonometric polynomial, homogeneous in the chosen degree.

We will never know exactly how Ramanujan found K1, K2, and K3, but his notebooks do evidence a propensity for exhaustive searches. In any case, the statement is not lacking insight, but certainly needs more rigor. We can do incrementally better by using Creative

H dφ Telescoping to print and verify Table 3.1. Here, right factors are written as I(α) = 1−αΦ and given alongside an annihilator such that I(α) = 0. When annihilator determines A A◦ A a hypergeometric I(α), the coeﬃcient recursion can be found by the Frobenius method. We will now give a brief, easy example of how this works in practice. The most simple choice, Φ = 4 sin(φ)2, recalls earlier calculations18 of E(α) and K(α).

H dφ The integral function I(α) = 1−4α sin(φ)2 has no square root, so details work out with even less eﬀort. The identity,

dI 1 2 cos(φ) sin(φ) ∂φΞ = 2 (1 4α)∂α ∂φ = 0. A ◦ dφ − − − ◦ 1 4α sin(φ)2 − 1 4α sin(φ)2 − − utilizes certificate Ξ to prove that almost annihilates dI . Again, exact differentials A dφ integrate to zero on a closed contour, thus completely annihilates I(α), or I(α) = 0. A A ◦ When solving for a coefficient recursion, it is also useful to view as a coefficient matrix, A 18Cf. [1] sec. 1-2. In fact, up to scale of α, the same matrix invariants can be used again. 87 Table 3.1.: Integral series I(α) = H dφ = P∞ H (αΦ)ndφ must satisfy I(α) = 0. 1−αΦ n=0 A ◦ s Φ = = I(z) pFq? A 2 2 2 P Q 2 (1 4α)∂α A000984 yes − − 2 2 2 2 I2Q same as for Φ = P Q A000984 yes 4 2 4 Q 6 (1 64)α∂α 2α(1 16α)∂ A001448 yes − − − − α 2 2 2 4 (Q4) same as for Φ = P Q A000984 yes 3 2 2 6 (PQ ) 30 3(1 452α)∂α 8α(4 243α)∂ A211419 yes − − − − α 2 3 (cont.) 16α (1 27α)∂ − − α 2 2 2 3 P2P 2(1 + 4α) (1 32α 112α 160α )∂α A288470 no − − − − 2 (cont.) 2α(1 + α)(1 + 4α)(1 8α)∂ − − α 2 3 P3P 12α(4 + 9α) 2(1 13α 84α 270α )∂α A092765 no − − − − 2 (cont.) α(1 4α)(1 + 6α)(4 + 9α)∂ − − α 2 2 (P3Q) 12(10 126α + 729α ) nAn no − 2 3 (cont.) 6(5 1093α + 7308α 13122α )∂α − − − 2 3 2 (cont.) α(320 11043α + 85806α 69984α )∂ − − − α 2 2 3 (cont.) 2α (5 54α)(16 207α + 108α )∂ − − − α 2 2 2 3 (Q3) same as for Φ = P Q A000984 yes 2 2 2 3 (I2Q) same as for Φ = P Q A000984 yes 2 2 2 4 (PQ ) 24 6(1 136α)∂α 18α(3 64α)∂ A005810 yes − − − − α 2 3 (cont.) α (27 256α)∂ − − α 3 2 2 6 (Q ) 40 2(1 904α)∂α 18α(1 144α)∂ A066802 yes − − − − α 2 3 (cont.) 9α (1 64α)∂ − − α 2 2 (P2Q) 24(1 16α) A005721 no − 2 3 (cont.) 6(1 248α + 2688α 9216α )∂α − − − 2 2 (cont.) 6α(1 16α)(9 276α + 512α )∂ − − − α 2 2 3 (cont.) α (1 16α) (27 32α)∂ − − − α 4 4 I2Q same as for Φ = Q A001448 yes 2 2 2 2 3 I2P Q same as for Φ = P Q A000984 yes 5 2 2 (P Q) 7560 30(3 184504α)∂α 30α(459 1076000α)∂ nAn yes − − − − α 2 3 3 4 (cont.) 423α (99 80000α)∂ 32α (729 312500α)∂ − − α − − α 4 5 (cont.) 4α (729 200000α)∂ − − α 2 2 2 2 6 (P2Q2) same as for Φ = (PQ ) A005810 yes ...... 1 2 2 Pn = 2 cos(nφ),Qn = 2 sin(nφ),P = P1,Q = Q1,I2 = 4 (P + Q ) = 1. See Appendix A. 88 2 −1 , where powers of α increase by row, and powers of ∂α increase by column. The A ∼ 0 4 matrix encoding of allows algorithmic streamlining of the Frobenius method, A (n + 1 ) I 1 an+1 (2, 4) (1, n) 2 dφ 2 , = · = 4 = I(α) = 2 = 2F1 · 4α . an (1) (1 + n) (n + 1) ⇒ 1 4α sin(φ) · − ·

In case it is not already clear, dots in the arguments of a 2F1 function indicate canceled parameters, so the evaluation is more precisely to a 1F0 function. The main virtue of Table 3.1 is systematism. Rather than searching by intuition through the space of valid Φ, we develop a comprehensive list of monomials, and use the diagnostic

ExpToODE to distinguish cases. We include any monomial of the variables Qn = 2 sin(nφ),

1 2 2 Pn = 2 cos(nφ), and I2 = 4 (P + Q ) = 1, which satisfies the degree constraint that d = P subscript exponent, with d = 4 or d = 3, and summing over all multiplicands. For × the quartic case, we find only 6 distinct cases, and only 4 distinct cases for cubic functions. These appear in the first and second divisions of Table 3.1. Perturbations Φ appear with an extra square when odd powers would otherwise integrate to zero. This is true for all cubic perturbations, and also for a few of the quartic perturbations, as we have already seen. Given the data of Table 3.1, matrix encodings of each annihilator can be inspected to determine whether or not I(α) is a hypergeometric function. If the matrix has non-zero values only on the central diagonal and the first upper diagonal, then it is hypergeometric and gets a ”yes” in the last column. Otherwise if the matrix form of contains non-zero A values on k > 2 diagonals, the Frobenius recursion will relate an, an+1, . . . , an+k−1, so it cannot be hypergeometric.19 We should not be too surprised to find a few more interesting

models for the series called by Ramanujan K1, K2 and K3. However it is somewhat amazing that all ”yes” hits lead to one of these periods. The asymmetric alternatives are also worth a look. Reading down the table, we ﬁrst ﬁnd

2 2 the pertubation Φ = I2Q = Q , and can recognize the corresponding Hamiltonian function,

2 2 1 2 2Hϕ = (p + q )(1 q ), as the algebraic form that describes simple pendulum libration. − 4 19There is a sometimes a caveat about ”minimal telescopers”, but all ”no” cases of Table 3.1 have been double checked for the minimal property by guess and check, and by referencing with OEIS. 89 √ The ﬁrst unknown is Φ = (PQ3)2, or in the full notation, 2H00 = p2 + q2 4 3 pq3. The 6 − 9 h 1 5 i 2 6 , 6 2 period function, T6(α ) = 2π 2F1 1 α , follows from the Hadamard decomposition,

1 , 5 1 , 3 1 , 1 , 5 F 6 6 α2 = F 4 4 α2 ? F 6 2 6 α2 . 2 1 2 1 1 3 2 1 3 1 2 4 , 4 Comparison with the similar cubic period reveals more than dancing parameters,

n 4n X 6n5n k 1 3n6n − − = . 2n k n k n 3n k=0 − a seemingly improbable, nonetheless true, binomial identity! Another way to prove the √ identity is to observe that the shear transformation p p + 2 3 q3 takes quartic H00 to → 9 6 sextic 2H0 = p2 + q2 4 q6. All such shears have Jacobian determinant 1, so they are 6 − 27 0 canonical transformations, thus preserve periods of oscillation. As it must, H6 also follows from Φ = (Q3)2 applied to the general sextic form. Another similar example for signature 4 is found from H00 = p2 + q2 pq2, by applying the shear p p + 1 p2. This canonical 4 − → 2 3n4n 2n4n transformation again obtains H4, thus n n = n 2n , another unexpected identity! The only cases left are those marked as s = 3 in Table 3.1. Applying an Abel-Wick √ √ 2 2 2 3 rotation, p ip, q 3 q to 1 H3 obtains He3 = (3p + q )(1 9 q). Up to a → → − − √ − 2 2 2 3 factor √3 on the frequency scale ω0, He3 (p + q )(1 q), what we obtain by applying ≈ − 9 Φ = I2Q to the general cubic form. Isoperiodicity follows from the fact that α 1 α acts → − 2 invariantly on the annihilator s. The last essential case is Φ = Q , with Hamiltonian form, A 3 0 2 2 4 3 2 2 0 2 2H = p + q (q 3p q) . By now, the derivation H and its period function T3(α ) 3 − 27 − 3 should be quite obvious. If not, we assert that sextic anharmonic oscillators of the form α = 2H(λ, φ) = 2λ λ3Φ are measured by the period function, − ∞ I 1 2 I X 1 3n 2n n 3 , 3 2 dφ T (α) = α Φ dφ = 2F1 α ? . 8n n 1 1 27 α2Φ n=0 2 − 32

3 Consequently, when a cubic function α = 2H = 2λ λ 2 Φ has period T (α), the corresponding − sextic function α = 2H = 2λ λ3Φ2 has period T (α2). − Having done so much more analysis, we can now strengthen the earlier theorem:

90 h 1 1 2 5 i 6 , 3 , 3 , 6 4 Table 3.2.: Sextic Hamiltonians and their periods, T (α) = 4F3 1 1 3 α ?I(α). 4 , 2 , 4

α = 2H = I(α) = T (α) =

1 1 2 5 8 10 h 1 ,· i h , , , i p2 + q2 (p5q p3q3 + q5p) 2π F 2 α4 2π F 6 3 3 6 α4 9 3 2 1 · 4 3 1 , 3 ,1 − − 4 4 √ 1 1 3 4 3 h , , i h 1 , 5 i p2 + q2 pq(p2 q2)2 2π F 4 2 4 α4 2π F 6 6 α4 9 3 2 1 , 2 2 1 1 − − 3 3 h 1 , 1 , 5 i h 1 , 1 , 5 , 5 i p2 + q2 4 (p2 q2)3 2π F 6 2 6 α4 2π F 6 6 6 6 α4 27 3 2 1 , 2 4 3 1 , 3 ,1 − − 3 3 4 4 √ h 1 , 1 , 5 i h 1 , 1 , 1 , 2 , 5 , 5 i p2 + q21 64 3 pq3 2π F 6 2 6 α4 2π F 6 6 3 3 6 6 α4 243 3 2 1 , 3 6 5 1 , 1 , 3 , 3 ,1 − 4 4 4 4 4 4 √ h 1 , 3 , 1 , 7 , 9 i h 1 , 3 , 7 , 9 i p2 + q2 32 5 pq5 2π F 10 10 2 10 10 α4 2π F 10 10 10 10 α4 125 5 4 1 , 1 , 2 , 5 4 3 1 , 3 ,1 − 6 3 3 6 4 4

3 2 Assuming either form, α = 2H = 2λ λ 2 Φ or α = 2H = 2λ λ Φ, a valid − − choice of monomial Φ determines a hypergeometric period T (α) if and only if

s T (α) = 0 for s = 2, 3, 4 or 6. Canonical models Hs all have elliptic level A ◦ curves s(α) for α (0, 1). Except for s = 2, each Hs has a higher-genus analog, C ∈ 0 and a canonical set of Hs for s = 3, 4, 6 can be chosen to maximize symmetry.

Take a step back and think about what the strengthened result says. Not only is it possible to ﬁnd Ramanujan’s set K1,K2, and K3 by searching a space of geometric models; if we

place mild constraints on the search space, we will only ﬁnd the set K1,K2, and K3! In that sense Ramanujan’s presentation is complete if not comprehensive. This curious case of good insight blurs the line between fortune and genius, but not between apathy and work. Don’t forget, Ramanujan did not have a computer to generate and check results! So long as we don’t pay too much attention to the particulars of series reversion, there is

0 0 no reason to cease the search at quartic degree. Given that H3 and H6 are sextic functions, it is natural to wonder: what happens generally for quintics and higher? Using ExpToODE we are able to perform an exhaustive search up to octic degree, over 200 choices of Φ. Most frequently the annihilators are not hypergeometric, but we also found many hypergeometric cases, including those of Table 3.2. In the subset of hypergeometric search results, we do not 91 n h a,b ni s−1 ﬁnd any surprising examples where T (α ) = 2F1 α with (a, b) = (s, ) or c = 1. For c 6 s 6 4 s = 6, the search uncovers another sextic model, though with period T6(α ). In a narrow

2 2 64 4 6 4 search above d = 8, we ﬁnd α = 2H = p + q p q , again with period T6(α ). − 27 The two extra occurrences of T6(α) both entail freak-accident parameter cancellations. More commonly, Hadamard products involve little-to-no cancellation. For d = 7 or d = 8 it

is already possible to ﬁnd periods of the form 10F8 or 10F9. The search results show a clear average pattern: as degree increases, complexity of the period function increases, and simple examples become sparsely distributed, if at all. We could continue to make observations and hypotheses, but again these would depend on unproven assertions. To prove claims more surely, we will now delve deeper into the algorithmic theory of Creative Telescoping.

3.3. Diagnostic Algorithms

Creative Telescoping is an algorithmic theory that aims to assist in the process of redeﬁning functions according to the ordinary diﬀerential equations they satisfy. We have already

made use of ExpToODE, with input/output map I(α) . Soon we will introduce two more → A algorithms, hyperellipticToODE and DihedralToODE. Despite diﬀering input domains, all three algorithms follow the same pseudocode, as written in Alg. 1. This pseudocode is a template for a range of C.T. algorithms that rely on degree-bounded Hermite-Ostrogradsky reduction [22]. Such algorithms are the simplest possible, while still meeting minimum rigor. They make a good starting place for curious beginners. Hermite-Ostrogradsky reduction is essentially limited to univariate cases; however, for the

H dI sake of versatility, it is useful to assume a chain-rule structure. The integral I(α) = dt dt requires a period T such that t t + T leaves dI invariant, so most likely, t will not make → dt a good basis for reductions. We need an algebraic variable z(α, t). A good choice for z

is often determined by inspection of ρ, for typically ρ Q[[i, α, z]]. In fact, the algorithm ∈ Pd n assumes ρ is a degree-d polynomial of z, i.e. ρ = n=0 cnz , and the cn themselves are usually rational functions or polynomials in the variable α. Subsequently we must also

92 Algorithm 1. Simple Creative Telescoping via Hermite-Ostrogradsky reduction. Input: An integrand dI α, z(α, t) , denominator ρ, and diﬀerentials ∂αz, and ∂tz. dI Output: Annihilator and certiﬁcate Ξ such that dt ∂tΞ = 0. A A ◦ − 8D 1: function EasyCT(dI,ρ,∂αz,∂tz): #(Ha, ha, ha! Have Fun!)> 2: d Deg(ρ); ∆ Deg(∂tz) 1; ← Pd+∆−1 ←n Pd−1− n 3: u unz ; v vnz ; ← n=0 ← n=0 4: G CoefficientMatrix(ρu (∂tz)(∂zρ)v, 2d + ∆); ← − 5: if Det(G) = 0 then return ”Error: no inverse for G.”; −1 ˙ 6: else (U, V) decomposition of G ; V (∂tz)∂zV; ←dI ← 7: n 0; x0 dt ; x0 CoefficientVector(ρx0, d + ∆); ← ← ← T 8: A NullSpace( x0 ); ← { } 9: while Empty(A) do n n + 1; xn = 0; ← 10: xn ∂αxn−1 + (∂αz)∂zxn−1; ← 11: if !Reducible(ρ, xn, n) then 12: return ”Error: ρ/dI mismatch.”;

13: w = w0, w1, . . . , wn PartialFractions(xn, ρ); { } ← 14: for m = 0, 1, . . . , n do 15: wm CoefficientVector(wm, d + ∆); ← ˙ 16: xn xn + VectorReduce(U, V, V, m, wm, d 1); ← T − 17: A NullSpace( x0,..., xn ); ← { } n 18: Ξ Reap(); return A 1, ∂α, . . . , ∂ , A SubTotal(Ξ) ; ← ·{ α} · 19: function VectorReduce(U, V, V˙ , k, w, l): 20: while k > 0 do 21: Sow( 1 1, z, . . . , zl V w); kρk { }· · 22: w (U + 1 V˙ ) w; ← k · 23: k k 1; ← − 24: return w;

Pd n−1 Pd+∆ 0 n requireρ ˙ = (∂tz) n=0 ncnz = n=0 cnz , i.e. thatρ ˙ is a z-polynomial. We will discuss in detail the obvious examples z = eit and z(t) = q(t), as well as a non-obvious, more diﬃcult case, z(t) = λ(t). Since these three cases are all strongly related, it should help to start with an overview of Alg. 1 including what it does, why it works, and when it can possibly fail. The ﬁrst few lines 2-4 determine the dimensionality of the reduction process, but do not guarantee success. Once variables are chosen, a feasibility analysis needs to be performed

93 relative to the central statement of Hermite-Ostrogradsky reduction from w to [w],

[w] 1 1 w v = u v˙ = ∂t w = ρu ρv.˙ ρm − m ρm ρm+1 − mρm ⇐⇒ − That such a reduction exists is itself an improbable assumption. The variables u, v, and w must be polynomials in the variable z with consistent degrees, i.e. deg(w) = deg(u) + deg(ρ) and deg(w) = deg(v) + deg(ρ ˙). Polynomials u and v are treated as unknowns with many degrees of freedom, dof(v) = deg(v) + 1 and dof(u) = deg(u) + 1. Solving the degree bound, deg(w) = dof(u) + dof(v), we obtain that deg(u) = deg(ρ ˙) 1 and deg(v) = deg(ρ) 1. We − − can also state that deg(u) = deg(v ˙) = d+∆ 1 in terms of d = deg(ρ) and ∆ = deg(∂tz) 1. − − This restricts all functions w(z) with implicit deg(w) = 2d + ∆ 1 to have non-zero wn only − from n = 0 to n = d + ∆ 1. When all of these conditions are met, it is possible to proceed − to line 5 of Alg. 1, where the ﬁrst critical error could possibly occur.

The G matrix is constructed relative to spanning vectors z = 1, z, z2, . . . , z2d+∆ and { } uv = u0, u1, . . . , ud+∆−1, v0, v1, . . . , vd−1 . It encodes the essentials of the right hand side { } of w = ρu ρv˙ . Introducing another coefficient vector w such that w = z w, we can − · rewrite the reduction constraint as w = G uv. If and only if det(G) = 0, the linear · 6 equation is uniquely solveable. If det(G) = 0, an error is thrown on line 5 and Alg. 1 fails. For some choices of z and ρ, it is possible to prove that det(G) never equals to zero, and this marks an important distinction between proveable and effective algorithms. Proveable algorithms are preferable because they will always work, whereas effective algorithms can only be guaranteed case-by-case. Absolute rigor is never necessary, and as long as possible errors are well-understood, it is not always desirable20. Line 5 is a major milestone for the algorithm, and passing it strongly suggests that the a positive result will be obtained upon halting. To explain subsequent line 6, we must split apart vector uv to u and v, lengths d + ∆ and d respectively, while dropping zeros of w to obtain w0, another vector of length d + ∆. Then we can define matrices U and V such that u = U w0 and v = V w0. Given that uv = G−1 w, matrices U and V must be submatrices · · · 20Our view is that extra rigor must sometimes be sacrificed to explore more deeply. 94 of G−1, so the first part of line 6 is no more complex than matrix inversion. Once the matrix V is known, the function v(w) is known, as isv ˙(w), as is the matrix V˙ . Both U and V˙ are square matrices, so we can finally achieve the goal of writing Hermite-Ostrogradsky reduction

in terms of linear algebra, [w0] = (U+ 1 V˙ ) w0, as in line 22 of Alg. 1. The halting condition m · is already in sight, but there is at least one more signiﬁcant chance of failure. Lines 11-12 validate that ρ is well-chosen with regard to integrand dI. As the length of

x0 is d + ∆, we can anticipate needing to calculate no more than D = d + ∆ reductions,

before the set x0, x1,..., xD must contain at least one linear dependency. Depending on { } symmetry, the ﬁrst linear dependency may occur for D0 + 1 vectors, with D0 < D. For this reason, we compute and validate derivatives on the ﬂy. To reach n = D0 and exit the loop

after line 17, we must always ﬁnd, for each n D0, that ∂n dI = Pn wm . In other words, ≤ α dt m=0 ρm+1 n dI the partial fraction decomposition of ∂α dt must yield no more than n + 1 degree-bounded Pd+∆ k numerators wm = k=0 wm,kz . During each loop, if the check on line 11 passes, then the

numerators wm are calculated on line 13. The algorithm enters a second loop on line 14,

and calculates the coeﬃcients wm,k on line 15. A third loop is entered on line 20 via line 16, and fourth loops occur on lines 21 and 22 as matrix multiplication. When this process repeats without error, the algorithm approaches halting on success. Eventually, on line 17, a

linear dependence between the xn will be found and set to A. The loop breaks, and shortly thereafter, the function returns a positive result. The subfunction deﬁned on line 19 deserves a closer look. It manages the iterative part of

wm Hermite-Ostrogradsky reduction by taking a wm m+1 from its initial form to its minimal ∼ ρ wf form wf . Lines 21-22 involve 2m matrix multiplications, only m of which are strictly ∼ ρ necessary. The sow step on line 21 keeps track of partial certiﬁcates by hiding them in the memory. Later, on line 18, they can be retrieved and summed. When subtotaled by index

n and dotted with A, the partial certiﬁcates Ξ determine a total certiﬁcate Ξ such that

dI ∂tΞ = 0. If the reduction reaches large recursion depth, speed and memory usage A ◦ dt − can suﬀer, so sometimes we choose to run the algorithm without calculating certiﬁcates.

95 However, if there is any doubt or misunderstanding, having a certificate assures quality. In practice, we also care about the complexity of Alg. 1. How does it scale relative to the degree bound D? Naively, the answer is that Alg. 1 is polynomial-time, roughly (D3+ω), O where 1 < ω < 3 is the complexity cost of matrix multiplication. In many cases three loops are not necessary, and the figure without partial fraction decomposition (D2+ω) looks even O better21. The problem is that matrix multiplication generates larger and larger polynomial data as recursion depth of the k-loop increases. This causes slow-down during the reduction phase, and eventually makes nullspace calculation prohibitively slow. If 2 < ω0 < 3 is the complexity of finding a null vector, then (D1+ω0 ) (D2+ω), and we could expect this O ∼ O step to dominate time dynamics. In many calculations, indeed it does. Heuristically, the complexity of Alg. 1 is much better described by an exponential or super exponential figure. If necessary, more accurate estimates can follow from testing on random inputs. Of course, this requires an implementation for some particular choice of inputs. In the previous section, we have already revealed that functions sin(t) and cos(t) make a good basis for an application of Creative Telescoping. The choice z(t) = cos(t) + i sin(t) is even better because ∂tz = iz. If we allow negative powers of z by requiring that ρ Q[[i, α, z, z−1]]then we can represent sin(t) and cos(t) as 1 (z 1 ) and 1 (z+ 1 ) respectively. ∈ 2i − z 2 z When ρ is chosen as ρ = 1 αΦ with Φ Q[[i, z, z−1]], and when dI = 1 we can make slight − ∈ dt ρ edits to Alg. 1 and arrive at the algorithm ExpToODE. In fact, we need only make changes

Pd n ˜ to account for negative powers. Assuming that Φ = d˜ cnz then bideg(ρ) = (d, d). As ∆ = 0, we can set bideg(u) = bideg(v) = bideg(ρ), which implies bideg(w) = (2d,˜ 2d) and dof(u) + dof(v) = 2(d + d˜) + 1 when v has no constant term. Thus the reduction constraint w = ρu ρv˙ leads to a square matrix G of size 2(d + d˜) + 1. After line 5 the algorithm is − much the same, except that the degree bound is changed to D = d + d˜+ 1.

The form ρ = 1 αΦ is simple enough to prove that ExpToODE halts on success for any − valid input Φ. As in Fig. 3.4, decompose G by G = G0 + αGα, with both G0 and Gα

21Both complexity estimates are comparable to estimates for similar algorithms [19]. 96     LT " # " #     LT LT I G =  I  + α   , G = + α     UT UT UT

UT for upper triangle, LT for lower triangular.

Fig. 3.4.: Forms for G matrices: ExpToODE (left), HyperellipticToODE (right).

˜ members of Q[[i]]. Under its ﬁrst dof(u) = d + d + 1 columns, matrix G0 contains an identity

submatrix, I, and these are the only non-zero values of G0. Expanding the determinant by minors determines a bound that det(G) = det(D) + (αd+d˜+1), with D a G-submatrix on O the space complement to that of I. In the D subspace, G0 = 0, so D αGα. Matrix D ⊂ has only one non-zero diagonal, and it is possible to identify that det(D/α) = ddd˜d˜ab when Φ = az−d˜ + ... + bzd. Finally we obtain det(G) αd+d˜ or det(G) = 0, thus an inversion ∝ 6 error is never thrown. As for derivatives, they can be written naively in terms of the central

n dI 2n−2 Φn binomial coeﬃcients ∂α dt = n−1 ρn+1 . A valid partial fraction decomposition must always exist, and a mismatch error between dI and ρ is never thrown.

n dI n! Pn n (−1)n+k With a little more eﬀort, it is possible to obtain a closed form, ∂α dt = αn k=0 k ρk+1 , for the partial fraction expansion. This particular form is an amazing fortuity because it

1 allows elimination of the loop on line 14. Once we have computed the reduction of ρn , we do not need to calculate it again during subsequent iteration of the n loop. In the optimized

1 algorithm, each xn can be calculated as the reduction of ρn . Once a linear dependency is found, it must exist in any linear transformation x0 = M x so long as det(M) = 0. In · 6 n (−1)n+k this case, M is a lower triangular matrix with non-zero elements mn,k = n! k αn , so det(M) = 0 always holds true. The coeﬃcients A can then be found from the nullspace of x0 6 and returned as usual. This is a nice optimization that helps ExpToODE to be an extremely successful workhorse, as we have already seen in the preceding section.

The other easy choice, not so obvious, is that z(t) = q(t) with ∂tz(t) = p(t), where q(t) and

97 p(t) are solutions to Hamilton’s equations given a hyperelliptic form α = 2H = p2 + 2V (q),

Pd n H and V (q) = n=1 cnq . The period is not a single number, but a function T (α) = dt = H dq n dI n 1 (2n−1)!! 1 p , with derivatives ∂α dt = p ∂α p = (−2)n p2n , as is proven using the chain rule with 1 ∂αp = 2p . Even though p is not a polynomial of q we can choose ρ = p if we rewrite that, [w] 1 1 w v 2 = u ∂qv = ∂t w = ρ u ρv.˙ ρ2n − 2n + 1 ρ2n ρ2n+2 − (2n + 1)ρ2n+1 ⇐⇒ −

This statement is not as straightforward. After applying chain rule, ∂tv = (∂tq)∂qv = ρ∂qv, the factor ρ cancels with the extra +1 power in the denominator. The degree bounding behaves as if ∆ = 1, and all reductions must be carried out to the zeroth power of ρ. Also, − the denominators in lines 21-22 need to be changed to replace k with 2k + 1. Other than these few minor changes, the resulting algorithm HyperellipticToODE exactly follows the template of Alg. 1.

For essentially the same reason as before, HyperellipticToODE is also a rigorously proveable algorithm. However, in this case the decomposition G = G0 + αGα has its

d−1 identity submatrix in Gα, consequently det(G) = α + smaller powers of α. The case d = 1 is of no interest, and the case d = 2 always results in = ∂α, as it must. In the A quadratic case, the Riemann surface associated to 2H = p2 + aq + bq2 is the genus zero harmonic hyperboloid, with T (α) = T0 = constant. For d = 3 and d = 4, usually the level curves of H are elliptic and T (α) = 0 has two solutions, corresponding to orthogonal A ◦ periods along real and complex time dimensions. For d > 4, typically contains 2g+1 A terms, with genus g = (d 1)/2 > 1. These cases are the proper hyperelliptic curves, b − c whose Riemann surfaces admit g distinct real periods and g distinct complex periods. In this sense, the proof of algorithm HyperellipticToODE is also a reproof of Fuchs’s theorem on the periods of hyperelliptic curves[36], though with slightly diﬀerent conventions.

Last but not least, we have the case of DihedralToODE, a very eﬀective algorithm, but not entirely proveable. For inputs we assume a form,

d1 d2 X n X n m α = 2H = c1,nλ + c2,nλ λ 2 cos(mφ) = h1 + h2 cos(mφ), n=1 n=0 98 so chosen because it can easily be solved to eliminate φ dependence, cos(mφ) = α−h1 . h2 Relative to Alg. 1, the choice z = λ is somewhat surprising, but it follows follows from the observation that λ˙ 2 and λ¨ can both be written as λ polynomials,

2 2 ˙ 2 m ¨ m λ = − (α h1 h2)(α h1 + h2) and λ = (α h1)(∂λh1) + h2(∂λh2) . 4 − − − 4 − ˙ Similarly, the angular derivative φ = h2(∂λh1) + (α h1)(∂λh2) /(2h2) works out to be a − rational function of λ polynomials, and we can usually choose ρ = h2(∂λh1)+(α h1)(∂λh2) . − Again, the basic assumption of Hermite-Ostrogradsky reduction must be rewritten, ˙ [w] 1 ¨ ˙ 2 1 w λv ˙ 2 = u (λ + λ ∂qv) = ∂t w = ρu λ v(∂λρ). ρn − n ρn ρn+1 − nρn ⇐⇒ − dI n dI ˙ n ˙−1 As before dt = 1 and subsequent derivatives are calculated as ∂α dt = φ∂αφ via the chain ˙ −1 ˙ 2 rule with ∂αλ = (2φ) . Degree bounding is possible with d = deg(ρ) and ∆ = deg(λ ) 1. − Thereafter, the best practice is to check for errors while computing and to require post hoc

dI quality analysis on the outputs. When ∂tΞ = 0, minimum rigor is still achieved. A ◦ dt − Something very interesting happens with the I/O map of DihedralToODE. For a typical

input with d1 = m/2 , d2 = 0, and d = (m 2)/2 , the ouput has 2d + 1 terms. We are b c b − c A tempted to identify g0 = d = (m 2)/2 as the genus of the Riemann surfaces associated b − c with α = 2H. This would foolishly contradict old and well-known theorems, which typically predict genus as a quadratic function of degree. Instead, what appears to happen is that the choice of dihedral symmetry gives the Riemann surface unexpected isoperiodicities, thus the number of distinct periods is fewer than the number of distinct homology classes. Since the dimension of counts periods, it cannot yield the genus without a complementary symmetry A analysis. We do not necessarily need homology/cohomology to complete the proof of Section 3, but will continue working toward these advanced topics anyways.

3.4. Finishing the Proof

Recall that in Section 2, we treated the problem of series reversion as too diﬃcult, and simply accepted assertions for period function left factors. How can we be sure that such a 99 factorization even exists? Consider the general form α = 2H = 2λ λd/2Φ, or if you like, − 2 2 αΦ d−2 = 2x + xd/2 with x = λΦ d−2 . The formal solution is either,

∞ ∞ 2 1 X d−2 n 1 X d−2 n x = αΦ d−2 + c (α 2 Φ) or λ = α + c (α 2 Φ) , 2 n 2 n n=1 n=1

d−2 2 +kn d d +k(n−1) 1+jn d−2 2 +j(n−1) d−2 so that chosen exponents j = 2 and k = 1 satisfy α Φ = α Φ . d−2 Then we can solve either constraint on coeﬃcients of α 2 Φ to obtain cn as a function of cn−1

with terminal c0. It is possible, not guaranteed that the cn obey a hypergeometric recursion,

but that property is not necessary to proceed. We only need to understand that the cn do not depend on choice of Φ and that they do depend on a choice of d. This allows us to produce a factored from for the period function,

I I ∞ X d−2 n T (α) = 2∂αλdφ = an(α 2 Φ) dφ = Ld α ?Id α , n=0

H d−2 −1 with Id(α) = (1 α 2 Φ) dφ. For odd powers of d, odd powers of Φ integrate to zero, − thus T (α) always expands in whole powers of α. The right factor is analyzable via ExpToODE.

2 We have already seen that Φ = Q leads to hypergeometric Id(α), and we can also calculate the hypergeometric parameters of the I(α) arising from Φ = Q2n, for any n. This allows

2 2 d proof by concordance between ExpToODE and HyperellipticToODE. If 2H = p + q cdq − has hypergeometric T (α), then the left factor L must also be hypergeometric. We can

find its parameters easily by comparing the parameters of Id(α) and T (α). Admittedly, the strategy is convoluted, but it trades the difficulties of series reversion for a routine algorithmic calculation. Ultimately, the main reason to prefer this constructive, data driven proof is that it allows us to gain more familiarity with the algorithms of Section 3. Now we just turn on the computer, instantiate the algorithms, calculate a few differential equations, and list them in Table 3.3. They are all hypergeometric, and they should22 exactly match previously asserted forms. As an example to show how the Hadamard factoring

22Unfortunately when typing out long equations typos are often made. Write the author if you notice any! 100 Table 3.3.: Period analysis of a few simple hyperelliptic Hamiltonians.

2V (q) = = T (α) = s A √ 2 3 3 2 h 1 , 5 i q 5 36(1 2α)∂α 36α(1 α)∂ 2π 2F1 6 6 α 6 − 3 − − − − α 1 1 4 2 h 1 , 3 i q 3 16(1 2α)∂α 16α(1 α)∂ 2π 2F1 4 4 α 4 − 4 − − − − α 1 √ h 1 , 3 , 7 , 9 i 6 15 q5 1701 2000(10 207γ)∂ 2π F 10 10 10 10 α3 125 γ 4 3 1 , 2 ,1 − − − 3 3 3 4 (cont.) ...... 90000γ (1 γ)∂ − − − − γ 4 6 2 h 1 , 5 2i q 5 36(1 2β)∂β 36β(1 β)∂ 2π 2F1 6 6 α 6 − 27 − − − − β 1 √ h 1 , 3 , 5 , 9 , 11 , 13 i 50 35 q7 12065625 57624(3136 815625)∂ 2π F 14 14 14 14 14 14 α5 2401 6 5 1 , 2 , 3 , 4 ,1 − − − 5 5 5 5

(cont.) ...... − − − − 5 6 (cont.) 4705960000 (1 )∂ − − h 1 , 3 , 5 , 7 i 27 q8 945 256(32 675γ)∂ 2π F 8 8 8 8 α3 256 γ 4 3 1 , 2 ,1 − − − 3 3 3 4 (cont.) ...... 36864γ (1 γ)∂ − − − − γ Dots indicate dropped terms. Variables are β = α2, γ = α3, = α5.

actually works, we use the output of ExpToODE[sin(φ)10] to determine that

I 1 3 1 7 9 ∞ dφ 10 , 10 , 2 , 10 , 10 3 X 1 10n 3n I5(α) = = 2π 5F4 α = 2π α . 1 α3 sin(φ)10 1 , 2 , 3 , 4 210n 5n − 5 5 5 5 n=0 The complement 1 , 3 , 1 , 7 , 9 / 1 , 3 , 7 , 9 = 1 determines a lower parameter 1 to { 10 10 2 10 10 } { 10 10 10 10 } { 2 } 2 go along with 1 , 2 from T (α). The extra parameter 1 we get for free, which leaves the { 3 3 } upper parameters only. Since no cancellation occurs, they will just be the lower parameters of I5(α). Now better than assertion, for d = 5, we derive that

1 2 3 4 ∞ 2 3 n 5 , 5 , 5 , 5 3 X 2 3 5n 3n Ld(α) = 4F3 α = α . 1 , 1 , 2 55 2n 3 2 3 n=0

dn A similar explicit calculation proves for any odd d that an and for any even d ∝ 2n dn/2 that an . It would be nice to have a second proof (perhaps using series reversion) ∝ n 101 especially as calculating Annihilators becomes time intensive as d increases. However, it is no bother to do the work up to d = 8. We have done this work, so we can assure the wide results of Section 2. The outstanding case α = 2H = p2 + q2 64 p4q6 should not be forgotten. It involves − 27 a diﬃcult decic perturbation, which we will deal with circuitously. Again, start instead with a simple hyperelliptic decic, α = 2H = p2 + q2 256 q10. The computer tells us via − 2135 HyperellipticToODE that the period function is annihilated by

2 4 2 3 4 3 4 = 189 125(15 368δ)∂δ 125δ(335 1144δ)∂ 10 δ (5 8δ)∂ 10 δ (1 δ)∂ , A − − − − δ − − δ − − δ

with δ = α4, and we can check the certiﬁcate if doubt about persists23. The hypergeometric A parameters of T (α) are 1 , 3 , 7 , 9 upper and 1 , 3 , 1 lower. Meanwhile, the parameters { 10 10 10 10 } { 4 4 } of right factor I10 are the same as I5, so by cancellation, we calculate parameters for the

1 2 3 4 1 1 3 shared left factor L10. They are , , , upper and , , lower. According to ExpToODE { 5 5 5 5 } { 4 2 4 } 1 5 3 5n with Φ = (z+ ) z these parameters also determine the L10 expansion coeﬃcients an , z ∝ n up to choice of scale for α. Writing the right factor as,

∞ 4n I dφ X X 6n 4n I(α) = = ( 1)kα4n, 1 α4(210 cos(φ)6 sin(φ)4) n + k k − − n=0 k=0 allows us to uncover yet another unexpected binomial identity,

4n 3n6n 5n X 6n 4n = ( 1)k. n 3n n n + k k − k=0

2 2 64 4 6 4 This identity essentially explains why α = 2H = p + q p q should have period T6(α ). − 27 If there is any doubt about veracity, the identity can be double checked using Zielberger’s algorithm for hypergeometric summations. The general strategy of concordance is eﬀective though convoluted. We would like to have more direct proofs, especially for the seven cases depicted in Fig. 3.2 and Fig. 3.3. These cases divide neatly into two classes: simple hyperelliptic or simple dihedral. Table 3.3 already contains three annihilators with identiﬁable signature. This is no surprise, as

23The certiﬁcate sums terms up to q35 in its numerator, so a computer will probably be necessary. 102 Table 3.4.: Period analysis of a few simple dihedral Hamiltonians. m cm = T (α) = s A m/2 α = 2H = 2λ cmλ cos(mφ) − √ 4 6 2 h 1 , 2 i 3 2 9(1 2α)∂α 9α(1 α)∂ 2π 2F1 3 3 α 3 9 − − − − α 1 2 h 1 , 3 2i 4 1 3 16(1 2β)∂β 16β(1 β)∂ 2π 2F1 4 4 α 4 − − − − β 1 √ h 1 , 2 , 3 , 4 i 5 24 30 216 250(5 108γ)∂ 2π F 5 5 5 5 α3 125 γ 4 3 1 , 2 ,1 − − 3 3 3 4 (cont.) ...... 5625γ (1 γ)∂ − − − − γ h 1 , 1 , 2 , 5 i 6 32 40 9(27 680δ)∂ 2π F 6 3 3 6 α4 27 δ 4 3 1 , 3 ,1 − − 4 4 3 4 (cont.) ...... 1296δ (1 δ)∂ − − − − δ √ h 1 , 2 , 3 , 4 , 5 , 6 i 7 400 70 450000 8232(343 93750)∂ 2π F 7 7 7 7 7 7 α5 2401 6 5 1 , 2 , 3 , 4 ,1 − − 5 5 5 5

(cont.) ...... − − − − 5 6 (cont.) 73530625 (1 )∂ − − h 1 , 1 , 3 , 5 , 3 , 7 i 8 27 25515 160(1024 341901ζ)∂ 2π F 8 4 8 8 4 8 α6 16 ζ 6 5 1 , 1 , 2 , 5 ,1 − − 6 3 3 6

(cont.) ...... − − − − 5 6 (cont.) 5308416ζ (1 ζ)∂ − − ζ m α = 2H = 2λ cmλ 1 + cos(2mφ) − 1 2 h 1 , 1 i 2 1 4(1 2α)∂α 4α(1 α)∂ 2π 2F1 2 2 α 2 2 − − − − α 1 16 2 h 1 , 2 i 3 2 9(1 2β)∂β 9β(1 β)∂ 2π 2F1 3 3 α 3 27 − − − − β 1 h 1 , 1 , 1 , 3 i 4 27 27 8(16 351γ)∂ 2π F 4 2 2 4 α3 32 γ 4 3 1 , 2 ,1 − − 3 3 3 4 (cont.) ...... 576γ (1 γ)∂ − − − − γ h 1 , 2 , 3 , 4 i 5 4096 384 375(5 128δ)∂ 2π F 5 5 5 5 α4 3125 δ 4 3 1 , 3 ,1 − − 4 4 3 4 (cont.) ...... 10000δ (1 δ)∂ − − − − δ Dots indicate dropped terms. Variables are β = α2, γ = α3, δ = α4, = α5, ζ = α6. 103 2 2 2 Table 3.5.: Canonical models Hs for which s = (s 1) s (1 2α)∂α s α(1 α)∂ . A − − − − − α 2 2 2 2 h 1 , 1 i 2H2 = p + q p q T2(α) = 2π 2F1 2 2 α genus 1 − 1 2 2 4 1 3 2 h 1 , 2 i 2H3 = p + q ( ) 2 (q 3p q) T3(α) = 2π 2F1 3 3 α genus 1 − 27 − 1 2 2 1 4 h 1 , 3 i 2H4 = p + q q T4(α) = 2π 2F1 4 4 α genus 1 − 4 1 √ 2 2 2 3 3 h 1 , 5 i 2H6 = p + q q T6(α) = 2π 2F1 6 6 α genus 1 − 9 1 0 2 2 4 3 2 2 2 h 1 , 2 2i 2H = p + q (q 3p q) T3(α ) = 2π 2F1 3 3 α genus 4 3 − 27 − 1 0 2 2 1 2 2 2 2 2 2 h 1 , 3 2i 2H = p + q (p + q ) + 2p q T4(α ) = 2π 2F1 4 4 α genus 3 4 − 4 1 0 2 2 4 6 2 h 1 , 5 2i 2H = p + q q T6(α ) = 2π 2F1 6 6 α genus 2 6 − 27 1 2 0 2 2 2 2 2 Taking α β = α reduces = 4(s 1) s (1 3α )∂α s α(1 α )∂ s. → As − − − − − α → A

0 we already know that H6, H4, and H6 have the required hyperelliptic form. The remaining

0 0 H2, H3, H3 and H4 all have simple dihedral symmetry, so their period functions can be proven more directly using DihedralToODE. We instantiate an implementation, map across a small search space, and in Table 3.4 list the four hits alongside a few other negatives. Since

DihedralToODE is only an eﬀective algorithm, we also list certiﬁcates,

4√6λ5/2 sin(3φ) λ3 sin(4φ) λ3 sin(4φ) 32λ4 sin(6φ) Ξ = , Ξ0 = , Ξ = , and Ξ0 = , 3 (3α 2λ)3 4 4(α λ)3 2 8(α λ)3 3 3(3α 4λ)3 − − − − 0 0 for H2, H4, H2 and H3 respectively. The results of DihedralToODE listed in Table 3.5 can be checked against these certificates. By judicious use of the chain rule, we simply by calculate a zero value, s dt ∂tΞs = 0, where choice of either + or accounts for the A ◦ ± − possibility of a sign error (whether accidental or not, sign errors do sometimes happen). The other three results of HyeperellipticToODE involve longer certificates, here omitted. Those annihilation relations can also be checked post computation if necessary (cf. Appendix A). The last step of the proof involves verifying the genus figures quoted in Table 3.5, column three. According to well-known genus degree bounds, we expect g = (d 1)/2 for b − c hyperelliptic cases and g = 1 (d 1)(d 2) for dihedral cases. The upper bound is only 2 − − 104 0 0 0 met for H3, H4, H6, H4, and H6. The other models, H2, and H3, involve hidden symmetries, P 1 which require a corrective term ∆ = rn(rn + 1). The correction ∆ characterizes − n 2 behavior of singular points at infinity. As early as 1884, Max Noether gave a procedure for calculating ∆[74]. The same basic idea of counting genus as g = 1 (d 1)(d 2) + ∆ is 2 − − nowadays built into standard algorithms of algebraic geometry[1]. In this case, we do not

1 0 1 need to reinvent the wheel, so simply input 2Hs = 0 or 2H = 0 into Singular software, − 2 s − 2 and use the built-in genus function to output the correct integer [71]. In principle, this I/O

1 0 1 approach only tells us the genus of a curve s( ) or ( ). The domain of the oscillation disk C 2 Cs 2 is unobstructed by critical points, so it is safe to assume24 that g ( 1 ) = g (α) when C 2 C α (0, 1). The strong statement of Section 2 stands true. ∈ For most purposes, the first four rows of Table 3.5 are a good-enough starting place. If not, a next step into the territory of higher genus takes into consideration the models listed in the final three rows. We have already found more exotic species, and are not quite done with the task of cataloging. One glaring omission on our part is that we have yet to mention 2H = p2 + 12q2 + 8q3 36q4 48q5 16q6, a genus 2 hyperelliptic model with − − − period T (α) T3(α). It belongs to an infinite class of hyperelliptic models, indexed by ∝ 1 degree d = 3, 4, 5..., each having a potential Vd(q) = 1 d(q) , where d(q) stands for 4 − T T the dth Chebyshev polynomial of the first kind25. Using HyperellipticToODE it is possible

d2−4 2 to prove that d = 2 (1 2α)∂α α(1 α)∂ , say for the ﬁrst ten or twenty cases. A 4d − − − − α It would be interesting to determine a d-dependent form for certiﬁcates Ξd, but not strictly

necessary since one proof of d has already been given. This family of Chebyshev potentials A is truly amazing in terms of hidden symmetry, but it doesn’t yield any new cases with g = 1 after d = 4, nor does it yield any more cases of identiﬁable signature after d = 6.

0 0 One last construction is worth mentioning. Both H3 and H4 feature a factorizable separatrix that decomposes to a product of hyperbolas. Generalizing on this form and

performing analysis via DihedralToODE, we ﬁnd a few more interesting cases to take note

24Perhaps this point could use more rigor eventually, but presently we take it as common sense. 25 There are many diﬀerent kinds of Chebyshev polynomials, here we mean n cos(x) = cos(nx). T 105 s = 4 s = 6

Table 3.6.: A few more use cases for DihedralToODE. Input: 2H = Output: = s A 2 1 4 λ λ 1 cos(8φ) (4 15α) + 16α(2 3α)∂α 4 − 8 − − − 2 2 +16α (1 α)∂α √ − 2 2 3 3 λ λ cos(6φ) (9 32α) + 36α(2 3α)∂α 6 − 9 − − +36α2(1 α)∂2 − α 32 2 16 3 2 2λ λ + λ 9 cos(6φ) 8(18 25α) 9(27 104α + 75α )∂α − 27 729 − − − − 9α(1 α)(27 25α)∂2 − − − α of, as in Table 3.6 and depicted above. The ﬁrst two rows list Hamiltonians with identiﬁable signature. The corresponding curve geometries are not oscillation disks in the strict sense because leading term λ2 does not allow for a harmonic limit at the origin. Yet period functions still exist, and they can be written as T (α) = Ts(α)/√α for s = 4 or 6. In the third row, the annihilator is not hypergeometric, but it does have the special property that

2 ∂α(α(1 α)(27 25α)) = (27 104α + 75α ). Remarkably, this operator also shows up in − − − a diﬀerent combinatorial search performed by Bostan et al. [20]. Even more waits to be discovered and explained. It would be nice to develop a constructive theory for explaining all higher-genus models in terms of the genus 1 examples given in Table 3.5, but we cannot do so presently. Instead of digging deeper into questions of existence and equivalence, next we will return to the practical task of using period functions to calculate numerical values.

106 3.5. Periods and Solutions

Physicists recognize hyperelliptic Hamiltonians without any qualms because the assumed

1 2 form requires separability, with kinetic energy 2 p . Indeed, the simplest cases, H6 and H4, are textbook examples often found under the heading of anharmonic oscillation or perturbation theory [65]. As models for data, H6 and H4 work well in situations where a period varies linearly around a harmonic limit. In this context, local analysis of H6 and H4 generalizes the simple harmonic approximation to the simple anharmonic approximation. If the data is precise enough, more terms can be added to the potential V (q) until the corresponding function T (α) suﬃciently matches data26. Sometimes it may be possible to measure a period function over an entire oscillation disk. These circumstances motivate us to develop an entire theory for solving period functions. An important question is: what can we exclude? Our own prejudices may tempt us to discard any functions not conforming to the

1 2 seperable form H = 2 p + V (q). This would be a crime of oversimpliﬁcation and an

unnecessary handicap to analysis in general. Hamiltonian H2 is a perfect example in support of inclusiveness, because the corresponding oscillation disk exactly models the libration motion of a simple pendulum. The role of transformation theory can not be ignored. It allows us to change a seemingly malformed perturbing term, p2q2, into something more familiar, V (q) = sin(q)2. We have no catalogue of all such transforms, but expect more to be discovered soon. The point is that even though dihedral models look funny compared to their hyperelliptic counterparts, they may eventually facilitate analysis. As nature produces a variety of forms, so should we. When formulating a space of viable models for anharmonic oscillation, a permissive attitude allows us to include any oscillation disk that we can manage to integrate. Algorithms of the previous sections allow a great many integral period functions to be calculated as solutions to output ordinary diﬀerential equations. Instead of evaluating an integral every time we need a value of function T (α), we only need to calculate integrals

26Although, there is one caveat. Period function T (α) does not uniquely determine potential V (q) [65]. 107 for a small set of initial data, say T1 = T (α1),T2 = T (α2),...,Tn = T (αn). When n equals to the order of , the system of equations should be uniquely solvable for T (α) as a A proper function. Our preference is for piecewise series solutions, but completely numerical solutions also work.

To show how solutions are carried out in practice, we will now solve Ts(α) = ∂αSs(α) for each of s = 2, 3, 4 and 6. The four cases are similar enough to proceed with variable s, except when determining initial data. Recall that action function Ss(α) has a natural geometric interpretation as the area enclosed by curve s(α). For this reason, Ss(α) makes a better C starting place than Ts(α), but first we need to transform the period annihilator into an action annihilator. Assume that S S(α) = 0, then ∂α S S(α) = 0 can be rewritten as A ◦ ◦ A ◦ 2 2 T T (α) = 0, and the identity T = s determines that S = (s 1) s α(1 α)∂ . A ◦ A A A − − − α It is again hypergeometric, but the missing middle term causes lower parameter c to equal zero. The usual coefficient recursion would have f1 f0/0. This solution does not work, so ∝ 27 another must be found. In general, S Ss(α) = 0 is solved by , A ◦ 1 s−1 1 s−1 Z 1 s−1 −2 s , s s , s s , s Ss(α) = C1α 2F1 α C0α 2F1 α α 2F1 α dα 2 − 2 2 1 s−1 s 1 s 1 s , s X n = C0 1 + − α + C1 + C0 − log(α) α 2F1 α + Cn α . 2 s2 2 s2 2 n>1 The first line is perfectly valid, but may be doubted. The second line gives an Ansatz, which is already correct up to a set of undetermined coefficients. The condition,

X n 2 4 0 = S Ss(α) = anα = (s 1)(s s + 1)C0 2s C2 α A ◦ n>0 − − − 2 2 4 6 2 + (s 1) (5 5s + 8s )C0 + 12s (1 + s)(2s 1)C2 72s C3 α + ..., − − − − sets up a recursion on the coeﬃcients an. As the summation continues to higher powers of

α, the pattern continues, and the ﬁrst appearance of Cn always occurs in coeﬃcient an−1.

The system of equations can be solved sequentially to determine Cn C0, but they are just ∝ the expansion coeﬃcients of the integral on the preceding line.

27Recall [59] Sec. IV-V, or see again: Hypergeometric Function, Second-Order ODE Second Solution. 108 Table 3.7.: Initial data of reversed action functions.

s 0C = 1C = 2 4 4.2725887222397812 ... = 3 4 log(2) √ − − 2 − 3 9 3 4.1533165583656958 ... = 3 √31 + 2 log(3) √4 − − 4 4 8 2 4.0014346021854628 ... = 3 √21 + 4 log(2) 3 − − 4 6 18 3.7842127941220551 ... = 1 3 + 8 log(2) + 6 log(3) 5 − − 4

The expansion around α = 0 requires C0 = 0, for ∂α(α log(α)) = 1 + log(α), but T (α) is

ﬁnite valued at the origin. The harmonic limit T (0) = 2π also determines C1 = π, so we h 1 s−1 i s , s need only one initial condition rather than two. With the solution Ss(α) = πα 2F1 2 α , very slow convergence becomes a problem as α approaches 1. Fortunately, symmetry allows for a second expansion, which converges rapidly where the ﬁrst diverges and vice versa.

Annihilator S transforms invariantly by α α = 1 α, so we can write a reversed A → − expansion ) α ( s S around α = 1 by simply taking Ss(α) and reversing Cn nC and α α . → → The identity S ) α ( s S = 0 holds through reversal of α, thus the solution is correct up A ◦ to determination of 0C and 1C . These initial data depend on the behavior near α = 1, which in turn depends on the choice of s. The zeroth coeﬃcient is just the area enclosed by the separatrix, 0C = area s(1) . Determination of 0C requires case by case integration. C Curves 2(1) and 3(1) bound equilateral polygons, so the corresponding interior areas can C C be calculated proportional to squared side lengths. Separatrices 4(1) and 3(1) are not C C polygon boundaries, but the area can be calculated by 2 R p dq across the maximum width

of the oscillation disk. When 0C is known exactly, then 1C can be determined to arbitrary

1 precision by solving Ss(α) = ) α ( s S at the midpoint α = α = 2 . Table 3.7 collects the results of our numerical calculations.

1 The two solutions are equivalent, so they can be stitched together at α = α = 2 to form 1 a piecewise solution Ss(α), which equals Ss(α) when 0 α 2 , otherwise it equals ) α ( s S pw ≤ ≤ 1 when 2 < α 1. As a computable function, Ss(α) implicitly relies on a truncation parameter ≤ pw N, which controls how many terms are summed in both expansions. Choosing N = 60 is 109 S2 S3 S4 S6 7 2

5 2

3 2

1 2 α α α α 0 0 0 1 011 1

1 1 Fig. 3.5.: Root solving, Ss(αn) = (n + )Ss(1) with n = 0, 1, 2, and 3. pw 4 2 pw

Table 3.8.: Semiclassical quantum values for h = Ss(1) and N = 4, as in Fig. 3.5.

s α0 = α1 = α2 = α3 = 2 0.15592230 ... 0.44694844 ... 0.70571106 ... 0.92129983 ... 3 0.15232502 ... 0.43916833 ... 0.69785371 ... 0.91761719 ... 4 0.14788266 ... 0.42934933 ... 0.68764827 ... 0.91260116 ... 6 0.14176761 ... 0.41544690 ... 0.67263195 ... 0.90470726 ...

−16 good enough to guarantee double precision, according to the error bound s Ss(α) < 10 A ◦ pw when α [0, 1]. Precision can always be increased by retaining more summation terms, but ∈ don’t forget, a reconstruction of Ss(α) with more terms also requires recalculation of the pw

coeﬃcient 1C . This is only a slight annoyance, one we must live with for now. We still don’t

know of a reasonable method to obtain exact solutions for the four values 1C , but strongly suspect the guesses28 in Table 3.7 are correct. As error reaches a maximum at the boundary, a less conservative error estimate is obtained by subtracting the numerical value from the (assumed correct) exact value. When this is done for the N = 60 approximation, we can

−20 estimate the error of Ss(α) less than 10 on the entire oscillation disk. Having deﬁned a pw set of computable functions with error bounds, we can now begin to calculate.

Plots of functions Ss(α), as in Fig. 3.5, allow us to see with our eyes how similar the

28The computer program RIES accepts numerical values as input and returns probable closed forms as output. It is free software and available online at https://mrob.com/pub/ries/. 110 s = 2, 3, 4, 6.

Fig. 3.6.: Toric cross sections with real/complex contours in blue/green.

alternatives are. All four have the same small-amplitude limit Ss(α) πα, but begin to ≈ diverge from one another as α approaches 1. Maximum diﬀerence occurs at α = 1. An

2(4−18/5) 2 easy bound, (4+18/5) = 19 , says that, at most, the value of one action function diﬀers from the value of another by about 10%. In addition to the graphs, we also want to calculate a few special ”quantum values”29. In principle, the action functions could be inverted to

−1 ﬁnd energy α = Ss (S) as a function of action S. However, when only a few values αn are

needed, it is much more eﬃcient to simply root solve Ss(αn) hn = 0, say by Newton’s pw − method. Calculating the so-called quantum values requires special choice of a density N/h,

h 1 which in turn determines hn = (n + ) for n = 0, 1,...,N 1. In Figure 3.5 we set the N 2 − action scale as h = 0C , choose N = 4. The values of Table 3.8 then determine the blue curves of Fig. 3.2. In those graphs, the area contained by any central n = 0 blue curve

1 is 8 0C , as is the area between any n = 3 curve and the separatrix. The area between any 1 two consecutive blue curves equals to 4 0C . Thus the total area enclosed by the separatrix is 1 1 1 8 + 8 + 3 4 0C = 0C as necessary. The quantum theory will contribute more signiﬁcantly to followup articles. Before then, we have a few other classical calculations to discuss. Computable instances of the alternative period functions, allow us to calculate alternative nomes, the q’s mentioned in the introduction. Yet the q’s seem to stand for questions unanswered, and we are left to wonder. What should we make of them? How are they used

29It is left as an exercise: ﬁnd matrices with comparable eigenvalues, see also [59] Sec. VI. 111 in the theory of elliptic curves? They are not the q’s of Hamiltonian mechanics, rather,

iτ they are nomes, which conform to the general expression q = e , with τ = TI/TR, the ratio between real and complex periods. Thus far, we have only dealt with real periods, but for the set of basic Hs the real periods are the complex periods, up to a change of Harmonic frequency and inversion of the energy scale α 1 α. The annihilator s transforms → − A invariantly by α 1 α, so we can write TI = i(ωs/ωs)Ts(1 α). To obtain the frequency → − e − 30 ratios, simply apply an Abel-Wick rotation from Hs Hes, as in Fig. 3.6, and ﬁnd the → harmonic frequency around a circular point of Hes. For signatures s = 2, 3, 4, 6, we calculate the corresponding complex-time frequencies ωes as ωe2 = 2, ωe3 = √3, ωe4 = √2, and ωe6 = 1. By convention, ω = 1, then follows ω /ω = 1 csc(π/s), as quoted in the introduction. In s s es 2 the case s = 2, we have already shown how the nome q contributes to an exact solution using Harold Edwards’s alternative theory [27]. More work needs to be done using other q’s

to solve the time parameterization problem on other Hs, especially case s = 3.

3.6. A Few Binomial Series for π

A very special feature of Ss(α) and Ts(α) is that they satisfy a Legendre-style identity,

Ss(α)Ts( α ) + Ss( α )Ts(α) = 2π 0C . It follows from Ts(α) = 2∂αSs(α) and S Ss(α) = 0, A ◦ after taking either limit α 0 or α 1. First observe, → →

00 00 ∂α Ss(α)Ts( α ) + Ss( α )Ts(α) = Ss(α)S ( α ) + Ss( α )S (α) = 0, − s s

00 s−1 because solving either S Ss(α) = 0 or S Ss( α ) = 0 produces S (z) = 2 Ss(z) with A ◦ A ◦ s s α(1−α) either z = α or z = α . Next take the limit,

lim Ss(α)Ts( α ) + Ss( α )Ts(α) = constant lim α log(1 α) + 0C 2π = 2π 0C , α→0 × α→0 × − ×

30Refer back to [58] Sec. IV. A rotational axis must be chosen to intersect a hyperbolic point on the separatrix. 112 1 and the identity is already proven, no problem! Choosing α = α = 2 and dividing both sides 2 by 2π , we obtain a few summations for 0C /π,

∞ n 2 2 4 X X 1 1 2(n k) 2k = − , π 32n (k + 1) n k k n=0 k=0 − ∞ n 9√3 X X 1 1 3(n k)2(n k)3k2k = − − , 4π 54n (k + 1) n k n k k k n=0 k=0 − − ∞ n 8√2 X X 1 1 2(n k)4(n k)2k4k = − − , 3π 128n (k + 1) n k 2(n k) k 2k n=0 k=0 − − ∞ n 18 X X 1 1 3(n k)6(n k)3k6k = − − . 5π 864n (k + 1) n k 3(n k) k 3k n=0 k=0 − − After dropping denominators, the corresponding integer sequences,

s = 2 : 1, 6, 56, 620, 7512, 96208, 1279168, 17471448, 243509720, 3447792656,...

s = 3 : 1, 9, 138, 2550, 51840, 1116612, 24999408, 575368596, 13518747000,...

s = 4 : 1, 18, 632, 27300, 1306200, 66413424, 3515236032, 191434588488,...

s = 6 : 1, 90, 20280, 5798100, 1854085464, 632693421360, 225235329359040,...

are not found in OEIS as of June 16, 2020. These are not exactly what Ramanujan had in mind for Section 14 of [77], but they appear similar. Perhaps they are too straightforward or not rapidly convergent enough, but at least they are easy to derive. We would like to carry this idea farther, but simply do not have time or space. Interested readers are refered to Jonathan and Peter Borwein’s ”Pi and the AGM”[18].

3.7. Conclusion

Famously, G.N. Watson (1886-1965) compared Ramanujan with the Italian renaissance by saying,

Ramanujan’s formula gave me a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of the Capille Midicee and see before me the austere beauty of the four statues representing Day, Night, Evening, and Dawn, which Michelangelo has set over the tombs of the Medicis. 113 We don’t want to take away Watson’s imminent and lordly right to a personal opinion. This sort of high praise is appropriate, but is it another example of Eurocentric bias from a leader of an Anglocentric society? Even to this day, the question of how we should view Ramanujan and his work remains persistently diﬃcult. An antithesis is to compare with the temple architecture at Namakkal. This could be more appropriate considering the profound importance of place in Ramanujan’s life. Such a comparison misses the point of Watson’s appraisal, which is to say that Ramanujan became a solid contributor to Western culture. There is a middle way where all arguments are cast aside and instead—just listen to the music play! Then Ramanujan’s works can be described as a type of music that came into existence before it could even be properly heard. In Watson’s day the idea of a world harmony was only a promise, one made amidst the machinations of World War. Subsequent generations did bring the promise to life. Although Ravi Shankar and Zakir Hussain do not hail from Tamil Nadu, they are also natural Indian citizens who used their work to transcend national boundaries. They helped to invent the truly audible genre of world music. The rest is on records, tapes, CDs, DVDs and probably a few youtube videos31. The musicality of Ramanujan’s mathematics bears some relation to the prosody of earlier Indian scientists. First Pingala gave sayings for binomial coeﬃcients, then Madhava for trigonometric functions, and after Ramanujan, now there is even a saying for elliptic integrals.

Is Ramanujan’s famous assertion of alternatives K1, K2, and K3 a product of the East or of the West? And whose hearing is it meant to reach? The chosen language is not Sanskrit, nor Greek, nor Latin. Nor are alternatives K1, K2, and K3 otherwise hidden from anyone. Mahatma Ghandi and his followers won their engagements worldwide. The future is not the exclusive property of any trading company. Notions of class and caste are falling into disrepute, while exclusive rights to practice science continue to expire. It has never been easier to make genuine progress! The garden of Mysterious ideas is always full with more

31Listen also: West Meets East Volumes I & II, Ravi Shankar and Yehudi Menuhin; Passages, Ravi Shankar and Phillip Glass; Remember Shakti Live at Jazz a Vienne, Zakir Hussain, John McLaughlin et al. And here’s one more, Tala Matrix, Zakir Hussain, Bill Laswell, et al. 114 surprises waiting to be found. What else grows in this pleasant and peaceful space? Perhaps beetles and butterflies are practicing symbiosis with the flora? Poetics aside, our plan already anticipates the remaining examples E-G, as laid out in the Prospectus of the dissertation prelude [59]. In one last chapter we will continue to develop our walk-through guide to the systematic analysis of symmetric curve geometries. Science is a process of developing bias, and in the end, science will defeat itself unless the practice of branching out is maintained. One weakness of the present article is that we have focused mostly on hypergeometric cases in order to suggest that only four achieve a minimal form. This point is at once worthwhile and irrelevant. We should, and we will, spend more time analyzing geometries such as case 3 of Table 3.6. It is not a hypergeometric case, so the Hadamard factor analysis does not even apply! Yet the periods of this geometry, and of other similar geometries, are still differentiably-finite. That is, the period and action functions are the solutions of ordinary differentiable equations. The analysis is not much more difficult, and we can make more progress easily. In Section 5, Table 3.8 is really more profound than it seems. The semiclassical eigenvalues do more than enable harmonic proportions in drawn figures. In physics, they provide a bridge between the classical theory and the quantum theory. Similar values can be calculated by matrix methods, but tunneling between states also needs to be accounted for. This is one focus of the next chapter. This effort will build upon algorithms defined in this article, so we have already completed much of the work necessary. Physics and math, practice non-duality and the way is clear.

Acknowledgements

Opponents Neil Sloane and Joerg Arndt helped to improve rigor through email correspondence and on [seqfan] and [mathfun] mailing lists. Bill Gosper knew the phrase ”singular modulus”, leading to reference [97]. Peter Paule provided the author access to the RISC Mathematica implementation of Zeilberger’s algorithm for hypergeometric

115 summation, which is packaged with Koutschan’s Holonomic Functions. In applicable cases, these tools have been used to double check results obtained by EasyCT implementations. Calculations and drawings are the author’s own work (usually with the aid of Mathematica [54]).

116 4. Developing the Vibrational-Rotational Analogy (Using methods from Creative Telescoping)

In this culminating article, we take advantage of mathematical similarities between vibrations and rotations to develop a rigorous procedure for semiclassical quantization in either domain. The physical notion of precession is introduced classically, as it pertains to symmetric and asymmetric tops. Previously developed Creative Telescoping algorithms then help us to derive a closed form for the asymmetric top’s precessional period functions. After a brief discussion of Quantum Mechanics, we utilize hypergeometric functions to approximate quantum energy levels and tunnel splittings, starting with quartic and sextic vibrational double-wells. The function-theoretic method generalizes readily to the related case of the asymmetric top. Finally, following William Harter’s interest in symmetric centrifugal distortions, we apply the same method to two additional cases featuring Octahedral and Icosahedral symmetry.

4.1. History and Introduction

The present dissertation, now to be concluded, can be described as a story about one of Euler’s profound topics—the interplay between period functions and the ordinary differential equations they satisfy. Through its Figure 12 and Table IV, the dissertation Prelude [59] outlines a substantial program for building this theme toward the realm of practical applications. Subsequent chapter [58] develops the physical mathematics of phase plane geometry, with special attention paid to the simple pendulum. A connection to Niels Abel’s development of double periodicity is found through Harold Edwards’s presentation of elliptic curves and elliptic functions [27]. Thereafter, the story takes a turn to the East, where the influence of Euler’s research into elliptic integrals and hypergeometric functions is even seen in one of Ramanujan’s most lasting works [77]. Mysteries left behind by Ramanujan afford us an opportunity to develop a powerful, algorithmic analysis, which

117 extends beyond easier cases when necessary [60]. With Creative Telescoping algorithms in hand, we now begin to traverse through the vibrational-rotational analogy. Euler was also among the first to consider the general problem of rigid body motion, and returned to this problem throughout his life [55]. The famous Euler’s equations are obtained primitively in Decouverte d’un nouveau principe de Mecanique (1752); however, these differential equations are more easily recognizable when they appear in body-frame coordinates, as in Du mouvement de rotation des corps solides autour d’un axe variable (1765)1. A profound convergence of mathematical ideas follows about a century thereafter. Euler’s work on rigid body rotation ultimately meets Euler’s work on elliptic integrals, by way of Jacobi’s Sur la rotation d’un corps (1850)2. In modern times, the exact solution in terms of elliptic functions can be read from Landau’s textbook on classical mechanics [65]. Landau’s presentation is notoriously difficult, but worth the effort. The text makes clear a deep connection between simple pendulum libration and rigid body rotation through its usage of the complete elliptic integral K(m) in both contexts. Lev Davidovich Landau (1908-1968) plays a prolific role in history of physics, similar to that of his predecessor Euler. As Euler arrives just in time to dominate research in all fields of Classical Physics, so too does Landau arrive just in time to dominate all fields of Modern Physics. Legend says that Landau wrote most of ”Course on Modern Physics” mentally, while unfairly imprisoned by the NKVD3. Whether or not this is true, Landau’s comprehensive knowledge begot the theoretical minimum, an admission exam so difficult that, even in the fierce Soviet educational system, only the best one or two students per cohort could pass. The present analysis is not so exclusive. Other than the aforementioned Vol. I, Mechanics [65], Chapter VI, ”Motion of a rigid body”, the most relevant reading is Volume III, Quantum Mechanics [66], Chapter VII, ”The Quasi-Classical Case”. In the preceding chapter on the simple pendulum [58], we hinted that phase portraits,

1In the Euler Archive [28], these works are catalogued as entries E177 and E292 respectively. 2GDZ online: https://gdz.sub.uni-goettingen.de/id/PPN243919689 0039, page 303:293. 3”The Top-Secret Life of Lev Landau”, Scientific American: https://www.jstor.org/stable/24995874. 118 such as originally given by Sommerfeld and Ehrenfest (cf. [58] Fig. 3), would reemerge in the context of Quantum Mechanics. When we see the curves of a phase portrait, we know that we can take integrals along them, usually by a wide variety of techniques. Once we have integrals, what should we do with them? One motivation comes from the Wentzel-Kramers-Brillouin (WKB) approximation, where action integrals help to approximate quantum energy levels. A detailed explanation of the WKB approximation is given by Landau [66]. More recently, this technique is also described in Eric Heller’s The Semiclassical Way [52] and Mark Child’s Semiclassical Mechanics with Molecular Applications [24]. Not only do these books give a mutually supportive approach to molecular physics, also they are both inclusive of William Harter’s symmetry theory, which we will get to shortly. Before we move on to deal with Harter’s theory, we will mention a couple more recent articles, [8] and [63], of interest and relevance to theory of semiclassical integrals. Both articles make case-by-case progress on refining the WKB approximation to higher orders. Even if all-orders WKB is not necessary (here it is not), some important insight can still be found at the methodological level. The idea is to use so-called Picard-Fuchs equations to expedite the process of semiclassical quantization. The patronymic ”Picard-Fuchs” is a synonym for what we have called ”annihilators”, or what P. Lairez and A. Bostan call ”telescopers”. It is quite surprising (to us at least) that neither [8] nor [63] makes any mention of other related works in the field of symbolic computing, for example [64] and [19]. In the physics literature, too much attention has been paid to all-orders calculation, while rigor and practice at first-order have been largely neglected. We have already given a few basic Creative Telescoping algorithms in [60]. Our subsequent analysis, though still mostly theoretical, will include data examples, and comparative timing statistics. This should help to make the case for updated methodology more convincing. The ultimate goal of the dissertation is not only to bring together the more disparate examples of [59] Table IV, but also to use all examples as part of a learning curve that

119 approaches William Harter’s theory of molecular symmetry. Harter’s textbook Symmetry Principles of Spectroscopy and Dynamics (SPSD) [49] gives its own systematic development, which we do not feel the need to significantly revise. However well this resource treats the vibrational-rotational analogy from the angle of representation theory, it does not say much about semiclassical mechanics. Semiclassical quantization of Rotational Energy Surfaces is an important topic in Harter’s published works [44, 45, 46, 48, 50], and we will develop a more thorough explanation as to how these sort of calculations ought to be done. After viewing many of Harter’s intricate figure drawings, we are tempted to conclude that his picture of molecular rotation is already well-enough understood, but then again where is the complex time? The later content of [58] Sec. IV is neither superfluous nor irrelevant. When working with the WKB approximation, we do not restrict our attention to real-valued segments of the Hamiltonian energy surface. By using the Abel-Wick rotation to define complex time, we extend contour analysis to include both real and complex cycles. These cycles link together at branch points, allowing a topological approach to the vibrational-rotational analogy. The primary advantage to using topology is that we can derive faster, more rigorous procedures for approximating quantum energy levels and their splittings4. That is our bigger goal, and the dissertation’s point of conclusion. Our first major task in Section 2 is to reprove and extend Landau’s result for the precessional period function of the asymmetric top. Then we develop the vibrational-rotational analogy until the idea of real and complex period functions transfers smoothly between domains. After limited discussion of the Abel-Wick rotation, we proceed to automatically construct real and complex period and action functions, with as few geometric integrations as possible. Ultimately, this allows us to compare accuracy and timing statistics between level calculations using either approximate semiclassical methods or exact matrix methods. The semiclassical method is shown to be competitive enough to entice more interest and future research.

4This thesis statement was ﬁrst given in a presentation at the International Symposium on Molecular Spectroscopy 2019, http://hdl.handle.net/2142/104330. 120 4.2. Rigid Body Rotation

An ideal rigid body is a collection of mass elements into an extensive whole, which responds with zero strain to any applied stress. Experimentally, we only encounter rigid (or inextensible) bodies in minimally stressful conditions. Let us continue with the ﬁdget spinner (Cf. [58] Sec. V) as a primary example. However, in this analysis we retain all three extra masses, so that balance maintains. The ﬁdget spinner’s plastic body bends slightly under a strong applied force, but such a condition is not usual to operation. While spinning, the pendulous masses place a centrifugal force on the plastic body, but this force does not cause appreciable change in shape. Every rigid body is characterized by a total

mass m and three principal moments of inertia Ix¯, Iy¯, and Iz¯ [65]. The moment of inertia

RRR dm 2 2 around a principalz ¯-axis is calculated by a volumetric integral, Iz¯ = B dV (¯x +y ¯ )dV dm with the local mass density of body and (¯x, y,¯ z¯) Cartesian coordinates. If indeed Iz¯ is dV B RRR dm RRR dm principal, we also require B dV (¯xz¯)dV = B dV (¯yz¯)dV = 0, which holds trivially when body transforms invariantly by reﬂection through thex ¯y¯-plane. B When Iz¯ is the moment of inertia along the spinner’s principal axis of rotation, moments

Ix¯ and Iy¯ along the remaining two principal axes must satisfy Ix¯ = Iy¯, why? In principal

coordinates, the inertia tensor I is represented as a 3 3 matrix with Ix¯, Iy¯, and Iz¯ along × the diagonal, and zeroes elsewhere. A coordinate rotation R by 2π/3 radians around thez ¯ axis changes matrix elements by a similarity transform, I0 = R I RT , · ·

 √     √   √  1 3 1 3 1 3 0 Ix¯ 0 0 0 (Ix¯ + 3Iy¯) (Ix¯ Iy¯) 0 − 2 − 2     − 2 2   4 − 4 −   √     √   √  I0 =  3 1     3 1  =  3 1 .  2 2 0  0 Iy¯ 0   2 2 0  4 (Ix¯ Iy¯) 4 (3Ix¯ + Iy¯) 0   −  ·   · − −  − −          0 0 1 0 0 Iz¯ 0 0 1 0 0 Iz¯

Yet a ﬁdget spinner transforms invariantly when it rotates 2π/3 radians around thez ¯ axis.

0 Consequently we require I = I , which implies Ix¯ = Iy¯ as desired. Fidget spinners belong

to the class of symmetric tops, while glass marbles with Ix¯ = Iy¯ = Iz¯ belong to the class

of spherical tops, and any body with Ix¯ = Iy¯ = Iz¯ is classified as an asymmetric top. B 6 6 121 Compared to the wonderful dancing T-handle5 (an asymmetric top), the rotational behavior of the fidget spinner is relatively tame. In normal usage, the fidget spinner only rotates its body’sx ¯y¯-plane via the ball bearing, which connects to the fixed finger grips. Angular momentum distributes entirely to one vector component Jz¯ = Iz¯Ωz¯, with angular velocity Ωz¯ around thez ¯ axis. Generally, angular momentum is written as a vector quantity, J = I Ω. Looking along the axis of rotation in · the direction J, an observer would see the rigid body rotating clockwise (this convention is sometimes called the ”right hand rule”). In a typical spinner rotation experiment, a kick is given so that J = (0, 0,J0). Over time, friction and vibrational couplings cause ∂t Jz¯ < 0 | | and component Jz¯ decreases toward zero. As a system with only one coordinate degree of freedom, the fidget spinner’s one standard behavior is to lose energy and stop spinning. However, if we change our perspective and introduce external forces, the more interesting behavior of gyroscopic precession can be observed. So far we have implicitly chosen body-frame coordinates by defining thez ¯-axis relative to the spinner’s central bearing. The lab frame usually defines the z-axis parallel to the local gravitational field vector g, while x and y span a horizontal plane. When J is a general, body-frame angular momentum, the corresponding lab-frame angular momentum can be written J0 = R J, where R is a rotation matrix with, at most, three angular degrees of · freedom6. If a fidget spinner body lies parallel to the horizontal xy-plane, thez ¯-axis and the z-axis happen to align allowing coincidence of angular momentum vectors, J = J0, but

0 0 more commonly J = J . An applied force F can cause a nonzero ∂tJ along the direction 6 0 J F while leaving ∂tJ close to zero. These are the conditions for precession, in this case, ± × a circular motion of the J0 vector around its z-axis. The simplest way to qualitatively observe ﬁdget spinner precession is as follows. Start

0 the spinner with a counterclockwise kick so that initially Jz¯ > 0 while J = J . Balance the spinner on one ﬁnger, and tilt it so that J0 has a small component in the horizontal plane. It

5Youtube, ”Dancing T-handle in zero-g, HD”, https://www.youtube.com/watch?v=1n-HMSCDYtM. 6In many cases, two angles will suﬃce, Cf. [49] sections 3.1.C and 5.3.A. 122 g

z¯ ←−

y¯ x¯ z

x y Fig. 4.1.: A Fidget Spinner Top.

may be necessary to wiggle the ﬁnger slightly to drive precession. When precession happens, the horizontal component of J0 only rotates clockwise around the xy-plane! Similarly, if the spinner is kicked clockwise, precession of J0 only happens counterclockwise. The parity of this driven precession diﬀers from that of standard gyroscopic precession, which we describe next7. More quantitative results can be obtained by turning the spinner into a proper symmetrical top. This is done by attaching one end of a metal rod to the inside of the spinner’s bearing, while setting the opposite free end on a horizontal surface8, as in Fig. 4.1. When J spins up to a large enough initial value, the rod tries to trace out the surface of a cone. Eventually

J0 declines its altitude, and the top topples over. If losses happen relatively slowly, an experimentalist can measure the inclination-dependent precessional period, i.e. the time for

J0 to rotate once around the lab z axis. Theoretical analysis of precession is complicated in general, but it appears in most textbooks on classical mechanics9. An experimental comparison similar to [58] Fig. 19 would be nice, but for now we move on to discuss force-free

7The interested reader is invited to prove why parity should diﬀer. 8Youtube, ”Fidget Spinner Gyroscope Tutorial”, https://www.youtube.com/watch?v=Lp0PRaJM2BA. 9An undergraduate textbook is [94], graduates can use [65] or [29]. 123 (a, b, c) = (1, 2, 5)

z¯ y¯ x¯

Fig. 4.2.: Phase Sphere and RES.

precession. In abstract theory, deterministic motion of the angular momentum vector happens in a six-dimensional phase space spanned by coordinate variables J = (Jx¯,Jy¯,Jz¯) and their time ˙ ˙ ˙ ˙ derivatives ∂tJ = J = (Jx¯, Jy¯, Jz¯). Force-free motion obeys Newton’s ﬁrst law, and angular

2 2 2 2 momentum must conserve, J0 = Jx¯ + Jy¯ + Jz¯ = constant. An applied force is needed to aﬀect work, so total rotational energy must also conserve,

1 α = I Ω2 + I Ω2 + I Ω2 = H (J) = aJ 2 + bJ 2 + cJ 2 = constant. 2 x¯ x y¯ y z¯ z E x¯ y¯ z¯

In geometric terms, conservation of angular momentum restricts J to the surface of a phase sphere, while conservation of energy restricts J to the surface of an energy ellipsoid. Choosing

a time scale, we ﬁx J0 = 1. Assuming an asymmetric top with a < b < c, the energy ellipsoid then intersects the unit phase sphere in four sets of nested, closed contours,

n 3 o (α) = J = (Jx¯,Jy¯,Jz¯) R : 1 = J J, α = HE(J) , C ∈ ·

as depicted in the bottom-left of Fig. 4.2. Intermediary energy α = b locates a separatrix

curve (b), which crosses itself at Jy = 1. This separatrix bounds four oscillation disks—two C ± at high energy α (b, c], centered on the body z-axis; and two at low energy α [a, b), ∈ ∈ 124 centered on the body x-axis. The disks are paired symmetrically according to the invariant transformation J J, which is functionally equivalent to time reversal symmetry, t t. → − → − To distinguish between low and high energy, we take each curve (α), rescale its points by C J αJ, and draw a Rotational Energy Surface (RES), as in the right of Fig. 4.2. → By deﬁning the phase sphere and its curves (α), an analogy to phase plane Hamiltonian C mechanics begins to emerge10. Other than the separatrix, all contours are closed loops. Each non-singular (α) can be integrated to obtain a precessional period, which determines C the time it takes the J vector to make one turn around either thex ¯-axis orz ¯-axis of the body reference frame. Over an entire disk, precessional periods collect into a function, with a harmonic limit at either α = a or α = c and a divergence at α = b. Limiting properties of the precessional period functions are assured by the rotational equations of motion, J˙ = 1 J J H(J) = J H(J) . The cross product requires that J˙ 2 ∇ · × ∇ × ∇ is always a tangent vector pointing forward along the curve (α). Explicitly we write that, C ˙ ˙ ˙ ˙ J = (Jx¯, Jy¯, Jz¯) = 2(c b)Jy¯Jz¯, 2(a c)Jz¯Jx¯, 2(b a)Jx¯Jy¯ , − − − for the asymmetric top, which matches the result of Landau and Lifshitz [65] up to change ˙ of variables. In a limit where Jz¯ approaches 1, energy α approaches c, Jz¯ approaches 0, ± and curves (α) become increasingly circular. The equations of motion are solved by simple C p trigonometric functions with harmonic frequency ωz¯ = 2 (c a)(c b). Similarly, energy − − minima are achieved around either circular point J = ( 1, 0, 0), where harmonic precession ± p occurs with angular frequency ωx¯ = 2 (b a)(c a). − − The third limiting possibility is an approach to either hyperbolic point at J = (0, 1, 0). ± Locally, the equations of motion are solved by hyperbolic trigonometric functions. Hyperbolic points work the same way on the phase sphere as they do in the phase plane. When the J vector gets into a sharp corner, it slows down exponentially, and may even appear to stop momentarily. This is exactly what happens in the dancing T-Handle experiment, where we visually see the T-Handle ﬂipping back and forth, as in Fig. 4.3. Here we must be careful

10Refer back to our previous development of [58], and compare with [50]. 125 807 817 827 837 847 Source: https://www.youtube.com/watch?v=1n-HMSCDYtM t(frames)

Fig. 4.3.: Progress of the T-Handle over one half period.

with deﬁnitions. Angular momentum conserves in the lab frame, so precisely speaking, the lab-frame vector J0 remains frozen constant. Instead, the body-frame vector J precesses through the abstract space of the phase sphere, tracing out a curve (α), while the T-Handle C body tumbles through real tangible space. This force-free precession diﬀers from what we have seen previously. Recall that forced precession of a symmetric top through a gravitational

ﬁeld involves J0 changing with time, while J remains frozen constant. For precession not to be an ambiguous term, relevant external conditions should always be stated. In Chapter VI of Mechanics [65], Landau and Lifshitz give the exact solution of rigid body rotation in terms of the Jacobian elliptic functions. Appearance of elliptic functions betrays a deep similarity between rigid body rotation and simple pendulum libration. To the point of data analysis, such detailed analysis is again overkill. More tractably, the original teachers also give a period function11, s ! 1 (b a)(c α) TE a, b, c α = 2 K − − . (c b)(α a) (c b)(α a) − − − − Unfortunately, the textbook explanation is diﬃcult to follow, especially as we are not the chosen worthy successors of Landau’s theoretical minimum. No, the rest of us mere mortals can only take another approach, hopefully not as steep. One is found by transformation to the phase sphere’s natural action-angle coordinates. Let

p 2 p 2 Jx¯ = 1 J cos(φ) and Jy¯ = 1 J sin(φ), − z¯ − z¯ 11We have changed notation slightly and chosen a label E as in Euler. 126 ˙ then tan(φ) = Jy¯/Jx¯, and the angular velocity φ follows from the chain rule, ˙ ˙ ˙ ˙ Jx¯Jy¯ Jx¯Jy¯ Jx¯Jy¯ Jx¯Jy¯ φ˙ = cos(φ)2 − = − J 2 1 J 2 x¯ − z¯ = 2pc a cos(φ)2 b sin(φ)2pα a cos(φ)2 b sin(φ)2. − − − −

Thus we derive an integral deﬁnition,

I I z¯ dφ dφ T a, b, c α = = , E φ˙ 2pc a cos(φ)2 b sin(φ)2pα a cos(φ)2 b sin(φ)2 − − − − which looks foreboding, but ultimately fails to stymie our proofs. The road less traveled will take us for a walk around all diﬃculties apparent in the integrand. A more direct and preferable method exists for deriving angular velocity φ˙. Substituting

for Jx¯ and Jy¯ in H(J), we obtain

α = H(λ, φ) = a(1 λ2) cos(φ)2 + b(1 λ2) sin(φ)2 + cλ2, − −

in terms of action λ = Jz¯. Coordinates (λ, φ) are canonical action-angle coordinates because ˙ ˙ they satisfy Hamilton’s equations, (λ, φ) = ( ∂φH, ∂λH). Then we can rewrite the period − integral relative to an action integral,

I z¯ z¯ z¯ let SE a, b, c α = λdφ then TE a, b, c α = ∂αSE a, b, c α .

z¯ One advantage is found by the fact that action SE admits geometric description as the

solid angle between the curve (α) and the phase sphere’s equator where Jz¯ = 0. Yet the C z¯ dTE alternative route via SE must lead again to the same integrand dφ as written above. Instead of retreading, we input H(λ, φ) to DihedralToODE, and read the output,

2 2 bE = (Σ1 3α) 4(Σ2 2αΣ1 + 3α )∂α 4(α a)(α b)(α c)∂ , A − − − − − − − α

where Σ1 = a + b + c and Σ2 = ab + bc + ca.

In proof-checking mode, algorithm DihedralToODE outputs a t-dependent certiﬁcate

z¯ dTE function, and verifies bE post computation. In this case, knowing bE and , we can also A A dφ 127 calculate a φ-dependent certificate, (b a)pc a cos(φ)2 b sin(φ)2 sin(2φ) Ξ = − − − − , E 3/2 4 α a cos(φ)2 b sin(φ)2 − − z¯ dTE such that bE ∂φΞE = 0 as necessary for bE TE(α) = 0. At this point, Landau’s A ◦ dφ − A ◦ result can be verified by substituting the functional form into the annihilation relation and reducing terms. There is another interesting proof, which builds upon our previous work. As promised in the preceding article [58], we have more surprising transformation properties to reveal. Recall the square-symmetric Hamiltonian, 1 α = 2H() = p2 + q2 (p2 + q2)2 + p2q2 = 2λ λ2 + λ21 cos(4φ). − 4 − 2 − Using either ExpToODE or DihedralToODE, we derive the period annihilator,

2 2 b() = 2 3α + 3α 4(1 4α + 3α + 2α 3α )∂α A − − − − − 4α(1 α)(1 α + α)∂2 . − − − α Again, it is solved by either function, 2 α 2 α 1 TR(α, ) = K or TI(α, ) = K − . π√1 α α 1 π√α α − − c−a b−α Choosing = , annihilator b() is equivalent to bE, up to change in scale α . c−b A A → b−a According to permutation symmetry, we have not just one, but six amazing, unexpected

equivalence transformations! A subset of these maps from b() to bE identify that, A A x¯ 2π c α b a 2π α a b c TE(α) = TE c, b, a α = TI − , − = TR − , − , ωx¯ c a b c ωx¯ c a b a − − − − y¯ 2π b α c a 2π α b c a TE(α) = TE a, c, b α = TR − , − = TR − , − , ωy¯ b a c b ωy¯ c b b a − − − − z¯ 2π c α b a 2π α a b c TE(α) = TE a, b, c α = TR − , − = TI − , − , ωz¯ c a b c ωz¯ c a b a − − − − p where ωy¯ = 2i (c b)(b a). For one choice of parameters (a, b, c) = (1, 2, 5), Figure 4.4 − − depicts the three periods, real values in blue, complex values in green. Real-valued functions

x¯ z¯ TE(α) and TE(α) can be used to analyze the motion of a rigid rotor such as the dancing y¯ T-Handle. Complex-valued function TE(α) seems useless at ﬁrst, but plays an important role in subsequent Section 5. For now, we turn course, and ﬁnd another bridge to cross. 128 (a, b, c) = (1, 2, 5)

4π/ωy¯

4π/ωx¯

2π/ωy¯ = 4π/ωz¯

2π/ωx¯

2π/ωz¯ α 1 3 512

Fig. 4.4.: Precessional Periods.

4.3. Looking To Quantum

Pendulum libration, top precession, and other instances of classical dynamical motion are easily available to citizen scientists, while similar quantum behaviors can remain elusive and mysterious. The lightning beetle’s yellow-green ﬂashing, the glow of a Light Emmiting Diode (LED), the ﬂuorescence of a Mercury lightbulb or a Neon sign, the beam or the spot of a laser—these visible emission phenomena hint at the existence of quantum mechanics. Unfortunately, our eyes fail to distinguish pure and separate colors, so we need a tool like the pocket spectroscope12 to expand apparent colors over the set of pure colors, and over the entire electromagnetic spectrum. Locations, heights, widths and shapes of various intensity peaks in a spectrum tell us about the dynamic interactions of photons, electrons, atoms, and molecules. Although quantum mechanics is not limited to the study of one particular type of data, this brief introduction focuses on emission and absoroption spectra, primarily as a lead in to subsequent sections13.

The wave feature that we sometimes can see as pure color, either wavelength λ0 or frequency ν0, also determines the energy E0 of a single light quantum, called a photon.

12Or diﬀraction-grating glasses, available through https://www.rainbowsymphonystore.com/. 13For an easy introduction, see Haken and Wolf on atoms and molecules, [38] and [39]. At the graduate level [80] and [73] are high standards. For something diﬀerent try [76] or [51]. 129 Three photon state variables λ0, ν0, and E0 are doubly redundant because, λ0ν0 = c and

E0 = hν0. In the standard metrology, both c and h are exactly-known conversion factors, the speed of light and Planck’s constant respectively. Observing a Mercury lightbulb through a spectroscope, we can locate fairly sharp peaks at 436 nm (blue), 546 nm (green), and 579 nm (yellow). A lightning beetle flash can happen around the Mercury line at 546 nm, but the spectral peak will have a width on the order of 60 nm. How does the theory of quantum mechanics predict and explain these features? Mercury is an atom with many excitable electrons, while a firefly’s flash is the product of large biomolecules interacting. Electrons change their states and release photons to conserve energy. More truthful answers are not easy to find for such complicated systems. Rather than drowning in difficulties, instead we will continue with a more elemental example. The Hydrogen atom is a system of one positively-charged proton interacting with one negatively-charged electron via the fundamental electromagnetic force. Finding the Hydrogen atom’s energy levels is a standard and recommendable starter problem for serious students of quantum mechanics. Schrödinger’sequation admits infinitely many solutions

indexed by a quantum number n. Each solution pairs an electron wavefunction ψn with a

2 7 −1 frequency νn or a wavelength λn = n /R, where Rydberg’s constant is R 1.097 10 m . ≈ × Two distinct states ψn and ψm are geometric distribution functions in real space, and they do not overlap very well. Sometimes an electron transitions from state n to state m, where

1 1 n > m. Then the electron spontaneously emits a photon with frequency νnm = Rc 2 2 . n − m An emission spectrum contains lines for all allowable transitions νnm. Many photons must be captured, so the spectrum is typically measured from a large ensemble of Hydrogen atoms excited by an electric current. About four of the fundamental transitions fall into the visible part of the electromagnetic spectrum, as in Fig. 4.5. Amazingly, the naive prediction matches experiment to within one percent. Congratulations Schrödingerand company! Schrödinger’sequation can be used across the periodic table, where it helps to explain patterns in the orbital configuration of electrons, including valency. It even helps to explain

130 .2 .0410 .1434 656.3486 (nm)

Fig. 4.5.: Visible components from the emission spectrum of atomic Hydrogen. Source: https://en.wikipedia.org/wiki/Hydrogen the combination of atoms into chemical molecules, but not exactly. The idea that a molecule can take shape, vibrate, and rotate depends fundamentally on the Born-Oppenheimer approximation [16, 52]. Not only does this approximation say that certain types of molecular motion exist, it also allows calculation of these motions separate from one another. Relative to huge atomic nuclei, tiny electrons move freely and easily through space, not as point particles, but as fully quantum wavefunctions. Atomic nuclei can also move like waves, but they are so slow that their motion affects the electrons adiabatically. That is, electrons always have time to correct their own motions in order to stay in the same electronic state. Similarly, vibrational states are assumed to be protected from rotational mixing by significant scale difference. Consequently, the Schrödinger equation can be solved in three successive stages. Such calculations are difficult, but not impossible.

Molecular hydrogen (H2), hydrogen chloride (HCl), and molecular chlorine (Cl2) all have similar electronic ground states. Both atoms H and Cl have one free electron in an outer s orbital. A covalent bond is formed between H:H, H:Cl, or Cl:Cl by the pairing of two free electrons, with spins14 aligned anti-parallel. In practice, Schr¨odinger’s equation for the ground electronic state can be solved with nuclei frozen at variable distance q. A detailed account for H2 is given in [39] Chapter 4. The primary output of the calculation is a potential function V (q), which relates energy of the electronic ground state V to the internuclear distance q. When V (q) reaches a local minimum at q0, a possible bound state is declared, and the next stage of the Born-Oppenheimer approximation comes within reach. Assuming the nuclei unfrozen but moving very slowly, we can reuse the Schr¨odingerequation to calculate

14For a very deep perspective on spin, see also Tomonoga’s Story of Spin [95]. 131 energy levels: α0, α1, α2,... V (q) D

Fig. 4.6.: Morse Quantization.

quantum oscillation of the bond length q0. As long as we have a potential function V (q), quantum oscillation is very similar to classical oscillation, except for the quantization of allowable states to a ﬁnite, discretely-spaced set.

The Morse potential V (q) = D(1 e−a(q−q0))2, is often used when discussing diatomic binding. − Not only is the asymptotic behavior correct, the exact level frequencies are easy to calculate using the semi-classical technique of coherent phase integration15. The classical Hamiltonian

1 2 is written α = H(p, q) = 2m p + V (q), with action integral,

I Z q+ S(α) = p dq = 2 p2m(α V (q)) dq, q− −

where V (q−) = V (q+) = α.

1 Solution of the constraint S(αn) = (n + 2 )h yields the energies,

1/2 2 νa 2D νa 2 αn = S(αn) 2 S(αn) , 2π hνm − 8π hνm 2 with νa = ca and νm = mc /h.

15See S.M. Blinder’s Wolfram Demonstration, WKB Computations on Morse Potential. More explanation of WKB is given in the next section. 132 Table 4.1.: HCl Vibrations (cm−1)

νnm Morse Data Fit

ν10 2885 2886 2885

ν21 2758 2782 2780

ν32 2631 2679 2676

ν43 2504 2576 2572 Data and Fit from [9].

1 1 Dividing by h, we also get frequency levels νn = νa(n + ) ρD ρm(n + ) , in terms of 2 − 2 q 1 2D 1 νa dimensionless constants ρ = and ρ = 2 . The phase portrait in the lower D 2π hνm m 8π νm part of Fig. 4.6 shows contours associated to αn for the ﬁrst few n. For a molecule like HCl, data is easily available and parameter values well known. Hydrogen and 35-Chlorine have masses 1.0078 amu and 34.968853 amu respectively.

Together their reduced mass is 0.97957 amu or hνm 912.5 MeV. In [67], the authors ≈ −1 determine dissociation energy D 35748.2 cm 4.432 eV. This leaves νa undetermined, ≈ ≈ but a value can be read off the 0 1 absorption spectrum, readily available through → NIST Chemistry WebBook16 and/or Hyperphysics17. When confronting the data, we don’t find a single peak for the vibrational excitation. Instead, we find upwards of twenty peaks, divided into two separate sets by a gap of about 23 cm−1. For now let us ignore splitting due to rotational energy. Measuring to the center of the central gap, we determine

−1 that ν10 = ν1 ν0 = 2886 cm and solve for hνa 23.81 KeV and subsequently for − ≈ −5 −7 ρD = 1.569 10 and ρm = 3.304 10 . × × The fact ρD ρm indicates that the anharmonic second term of νn only contributes to level frequency as a small perturbation. Roughly, transition frequency νn+1 νn with n > 0 is − expected slightly less than ν10, as is found in the ﬁrst column of table 4.1. Comparison with data in the second column shows good accuracy, with percent diﬀerence increasing almost

16https://webbook.nist.gov/cgi/cbook.cgi?ID=C7647010&Mask=80#IR-Spec. 17http://hyperphysics.phy-astr.gsu.edu/hbase/molecule/vibrot.html. 133 linearly from 0.03% on the ﬁrst line to 2.8% on the fourth line. Authors of the experimental

−1 −1 paper give best-fit parameters ωe = νaρD 2988.9 cm and ωexe = νaρm 52.1 cm . ≈ ≈ The corresponding predictions, in column three of Table 4.1, never differ by more than 0.2%. We choose to emphasize that the best fit is inconsistent with dynamics in a Morse potential when dissociation energy and reduced mass are included as extra data. Actually, the Morse assumption does not yield 99.8% accurate predictions on the four lowest transitions, but the more strict figure 97% is not bad either. Nitrogen has three valence electrons in a p orbital with six total vacancies, so it combines naturally with three Hydrogen atoms to form Ammonia (NH3). This molecule is an important part of Feynman’s teaching in his Lectures Vol. III [31]. Modulo rotations and translations, configuration degrees of freedom number 6 = 4 3 6. Assuming C3v × − symmetry, we can reduce these 6 d.o.f. to only 218. One d.o.f. is a scalar stretching mode that simultaneously varies each N:H bond length. The other is a scalar bending mode, also called the umbrella mode, which fixes bond length and tilts inclination of the H atoms by an angle θ relative to a plane containing the N atom, as in Fig. 4.7. Since the Ammonia molecule has dihedral symmetry when θ = 0, the Born-Oppenheimer approximation is guaranteed to break down somewhere. Scalar vibrational modes might show a pseudo Jahn-Teller effect [16], but this does not seem to be a problem in Ammonia’s ground state. The maser experiment, as described by Feynman, takes advantage of a low-energy

−1 microwave splitting, which measures ∆ν0 = 0.791 cm . The vibrational absorption

± ∓ 0 1 also shows a relatively small splitting, so we can infer that ∆ν0 occurs on the → vibrational ground state, an umbrella mode. Inversion symmetry of the molecule’s umbrella

mode must carry over to the vibrational potential. For NH3 to reach a stable, pyramidal conﬁguration, the potential should be a double-well with local minima on either side of θ = 0. Fig. 4.7 depicts two simple candidates, a quartic V (q) = q2 + 1 q4 and a sextic − 4 V (q) = q2 + 4 q6. Either of these potential functions explain that spontaneous breaking of − 27

18 A more complete discussion of NH3 normal modes is given in PSDS [49]. 134 DATA Quartic V (q) Sextic V (q)

N inverted equilibrium

θ H H H ∓ ∓ ∓ 3 2 1 → → → ± ± ± 0 0 0 NH3 umbrella bending Compare with Table 4.2

Fig. 4.7.: Eﬀective potentials for Ammonia inversion in the umbrella mode.

−1 Table 4.2.:NH 3 Vibrations (cm ) Experimental Data Quartic Fit Sextic Fit n 0− n+ 0+ n− 0− n+ 0+ n− error 0− n+ 0+ n− error → → → → → → 1 932 968 859 904 7.24% 915 986 0.01% 2 1597 1882 1443 1758 8.01% 1485 1891 2.98% 3 2384 2896 2234 2747 5.65% 2418 3011 2.84% − Error calculated as percent diﬀerence between means. Data from [41].

dihedral symmetry at θ = 0 occurs due to the existence of energy minima at θ = θ0. An ± exact relation between potential coordinate q and the geometric coordinate θ is assumed to

exist, but need not be speciﬁed other than θ(q0) = θ0. Comparing Fig. 4.7 and Table 4.2, it is clear that the sextic potential yields better agreement with vibrational overtone data.

Our preferred analysis does not depend on the equilibrium angle θ0 0.386, but at ﬁrst ≈ ± −1 we do take barrier height B = V (0) V (θ0) 0.2193 eV (or B/h = νB 1769 cm ) − ≈ ≈ as a known absolute scale for calculated energy levels. The time independent Schr¨odinger

ˆ ± equation can be written as H ψn, = E ψn, . Eigenstates ψn, are labeled by a | ±i n | ±i | ±i vibrational quantum number n and a deﬁnite parity + or . The quantum Hamiltonian Hˆ − is obtained from the classical Hamiltonian 2H = p2 + V (q) by exchanging classical variables

for their quantum counterparts, i.e p √~Pˆ and q √~Qˆ. In the algebraic formulation → → of quantum mechanics, Pˆ and Qˆ are inﬁnite-dimensional matrices expanded over a complete 135 and orthonormal basis of states, call them φn . We choose φn as harmonic oscillator | i | i eigenfunctions and the matrix elements can then be written as19,

ˆ i φm P φn = i (√mδm−1,n √nδm,n−1), h | | i √2 − ˆ 1 φm Q φn = (√mδm−1,n + √nδm,n−1), h | | i √2 ˆ ˆ2 ˆ2 and φm N φn = φm P + Q φn = (2n + 1)δm,n, h | | i h | | i with δm,n = 1 if n = m and δm,n = 0 otherwise. The action scale ~ = h/(2π) controls density of states. Smaller values for ~ pack adjacent states more closely together, so action ~ is a good parameter for fitting spectra. Computable matrix representations of infinite-dimensional quantum variables can only be enumerated to high dimension, say 100 100 elements, but truncation error does not × significantly affect convergence of the lowest lying eigenvalues. Choosing either the quartic or sextic potential, we utilize matrix multiplication to construct Hˆ as a matrix. A value

± must be chosen for scale ~, then a diagonalization algorithm yields numerical eignvalues En . The lowest 4 or 8 eigenvalues can be read out in pairs. Per n, the symmetric state always

+ − exists at lower energy, i.e. En < En . We choose optimal values ~ = 0.2056 (quartic) and − + ~ = 0.1751 (sextic) by requiring agreement between predicted ∆ν0 = ν0 ν0 and the ﬁgure − quoted above. Then the eigenvalues of Hˆ predict the gross vibrational spectrum, as in Fig. 4.7 and Table 4.1. The better sextic potential achieves 97% accuracy overall, with 99.99%

1 + − + − agreement on the average value ν01 = (ν +ν ) (ν +ν ) of the fundamental excitation. 2 1 1 − 0 0 If instead we choose ~ to force agreement with the datum ν01/∆ν0 1202, we could expect ≈ to extract barrier height B quite accurately. During stage three of a Born-Oppenheimer calculation, we can ﬁnally address rotational sidebands as seen with the example of HCl (and with many, many others). So long as we only care about linear molecules, the answer is easy. Rotational excitation adds a small

19These definitions assume a commutator [Q,ˆ Pˆ] = Qˆ Pˆ Pˆ Qˆ = iI, with identity matrix I, as is obtained R ∗ ∗ · − · R ∗ from φn [ˆq, pˆ] φm = φnq( i∂qφm) + φni∂q(qφm) dq = i φnφmdq = i φm φn = iI. This choice of h | | i − h | i coordinates agrees with standard convention, after factoring out ~ = h/(2π). 136 perturbation νj = ν0 j(j+1), with integer quantum number j > 0. A photon carries one unit of angular momentum, so absorption must change the HCl molecule’s angular momentum by ∆j = 1. The frequency formula combined with the selection rule leads to equal spacing20 ± for rotational lines on either side of the missing Q branch, which does not appear because transitions with ∆j = 0 are disallowed. The sideband with ∆j = +1 is also called the R branch, while the P branch has ∆j = 1. Line amplitudes are determined by a combination − of state degeneracy and thermodynamics. Around ∆j = 0, peak height first increases due to hidden 2j + 1 degeneracy. As j increases, occupancy becomes exponentially suppressed due to thermodynamic difficulty of populating higher energy states. Thus peak heights far enough to left or right fall off to zero. Perhaps the best available reference for the theory of quantum rotational energy levels is Harter’s Principles of Symmetry, Dynamics, and Spectroscopy (PSDS)[49], which includes an explanation of Schwinger’s representation in terms of ladder operators. The textbook does not need to be rewritten, but we will state a few definitions for the purpose of internal completeness. Again, we promote continuous classical variables to discrete quantum variables by a mapping J ~Jˆ, and represent Jˆ as a set of three infinite dimensional matrices, with → elements,

j ˆ j 1 p p Jx¯ = δm,n−1 (j m)(j + m + 1) + δm−1,n (j n)(j + n + 1) , hm| |ni 2 − − j ˆ j i p p Jy¯ = δm,n−1 (j m)(j + m + 1) δm−1,n (j n)(j + n + 1) , hm| |ni −2 − − − j ˆ j Jz¯ = δm,nm, hm| |ni

21 ˆ ˆ ˆ where indices m and n range from j to j in integer steps . Matrices Jx¯, Jy¯, and Jz¯ are − all block diagonal, with each block of size (2j + 1) (2j + 1), and we have already broken × them down to ﬁnite parts. The subscriptsx ¯,y ¯,z ¯ indicate that Jˆ is a body frame variable.

20The HCl lines are not exactly equally spaced, because the HCl molecule distorts under centrifugal force. As with vibrations an anharmonic perturbation can be added to account for line shift. 21 As with Pˆ and Qˆ, we can check correctness by a commutation relation, [Ju,Jv] = uvwJw, with uvw the Levi-Civita tensor, uvw = 1 when uvw = xyz + cyclic permutations, uvw = 1 when uvw = zyx + − cyclic permutations, otherwise uvw = 0. Like it or not, prove it or no, that is a standard deﬁnition. 137 Presently we are not concerned with lab frame or its eigenstates, except for one important fact. In isotropic conditions, lab/body duality always multiplies an extra factor 2j + 1 to the total degeneracy of states (Cf. PSDS [49] Section 5.5.C, Fig. 5.5.6 and Fig. 5.5.7).

Linear molecules are an exceptional case where we do not really need the J matrices. They are assumed not to rotate around their bond axis, so body states have no degeneracy. The factor 2j + 1 apparent through data is due solely to the lab states, or equivalently, to the direction of the J vector in the lab frame. The J matrices are necessary for quantum tops—whether spherical, symmetric, or asymmetric. We will not repeat the discussion of PSDS, but refer readers again to critical Section 5.5.C. In subsequent Section 5 we will resume discussion of asymmetric rigid rotors and symmetric rotors with centrifugal distortion. To get there we need to take a ”semi-classical leap”. Emissions (or absorptions) due to electron state transitions happen at relatively high energy, sometimes in the visible, sometimes in the ultraviolet. Electric dipole absorptions (or emissions) due to vibration or rotation of a molecule happen at lower energies, often in the microwave. In the later circumstances, we can double check matrix calculations against semiclassical approximations. Semiclassical Mechanics sometimes gets talked down for its inexactness, but it will give us great intuition and extend our range of calculation tools.

4.4. The Semiclassical Leap

Now we will take a more serious look at the provenance of quantum theory by developing a relation between quantum energy eigenvalues and classical conserved energies. Schr¨odinger’s ˆ complex-valued wave equation, i~∂t Ψ = H(ˆp, qˆ) Ψ , is usually solved by a superposition of | i | i P∞ −iωnt states, Ψ = cne ψn , where ωn = En/~ = 2πνn. As in classical mechanics, | i n=0 | i the Hamiltonian Hˆ (ˆp, qˆ) is an energy function, with kinetic and potential terms. The ˆ general solution implies a time-independent equation, H(ˆp, qˆ) ψn = En ψn , which relates | i | i each eigenstate ψn to an eigenvalue ~ωn. It is natural to ask what a time-independent | i wavefunction ψn might look like in phase space. Doing so we immediately encounter | i 138 quantum weirdness. In phase space, the uncertainty principle, ∆p∆q ~/2, places a lower ≥ bound on the product of wavefunction widths,

Z ∞ Z ∞ 2 2 2 ∗ 2 ∗ ∆p = Ψ pˆ Ψ Ψ pˆ Ψ = ΨpΨp p dp ΨpΨp p dp , h | | i − h | | i −∞ − −∞ Z ∞ Z ∞ 2 2 2 ∗ 2 ∗ and ∆q = Ψ qˆ Ψ Ψ qˆ Ψ = ΨqΨq q dq ΨqΨq q dq . h | | i − h | | i −∞ − −∞

We are almost correct22 when interpreting the uncertainty principle as saying that the wavefunction Ψ exists deﬁnitely in only half the total dimension of phase space, for example | i as Ψq = q Ψ over position q or as Ψp = p Ψ over momentum p. In fact, wavefunctions h | i h | i Ψq and Ψp relate by Fourier transform,

Z ∞ Z ∞ 1 −ipq 1 ipq Ψp = Ψqe dq and Ψq = Ψpe dp (~ = 1). √2π −∞ √2π −∞

Thus the uncertainty principle is sometimes proven as a property of the Fourier transform. If any of this is unfamiliar to the reader, it may be worthwhile to peruse a few standard texts in order to recall basic deﬁnitions and notations. At a deeper level, we know that the diﬀerence between quantum position and momentum

follows from the commutation relation [ˆq, pˆ] = i (again with ~ = 1). Either allowable

assignment (ˆp, qˆ) ( i∂q, q) or (ˆp, qˆ) (p, i∂p) introduces a preferred axis. Similarly, → − → in the algebraic formulation, the statement [Q,ˆ Pˆ] = iI means that Qˆ and Pˆ cannot be simultaneously diagonalized. Then we usually have a Hamiltonian Hˆ such that [H,ˆ Pˆ] = 0 6 and [H,ˆ Qˆ] = 0, so Hˆ cannot be simultaneously diagonalized with either Pˆ or Qˆ. These 6 statements about the flexibility of quantum theory rely upon internal consistency. Whether by solving a differential equation, by diagonalization of a matrix Hˆ , or by another method, we should always end up calculating the same energy eigenstates. The appearance of complex phases in Schrödinger’sequation suggests a more direct question about phase space. What happens when position and momentum are allowed to take on complex values? We already asked this question, almost as a non sequitor, when

22 A method exists for lifting projections Ψp and Ψq to a density function ρ(p, q), which also obeys the uncertainty principle. See also [52] Ch. 9, shouldn’t we have more than three pages here? 139 analyzing phase plane geometries in two previous articles (or chapters) [58, 60]. The classical Hamiltonian H(p, q) introduces two constraints, say α = R(2H) and 0 = I(2H), which locate valid (p, q) C2 on a two-dimensional Riemannian (or Kleinian) surface, as in [58] Figures ∈ 10-13. Real-valued, classically allowed points (p, q) R2 determine a one-dimensional subset ∈ of the entire energy surface, which is completed by the remaining, classically forbidden points (p, q) C2/R2. In semiclassical mechanics these concepts allow us to ask pointed ∈ questions. For example, why do the nodes of a wavefunction form in classically-allowed regions? And why do wavefunctions dampen exponentially in classically forbidden regions? For relatively simple answers, we now turn to the famous Wentzel-Kramers-Brillouin (WKB) approximation, one of the earliest techniques of semiclassical mechanics [24, 52]. Suppose that somewhere in a molecule, a vibration happens along a Born-Oppenheimer

1 2 potential surface V (q). Adding in a kinetic term 2m p produces H(p, q) from V (q), and we can expand an oscillation disk around equilibrium conﬁguration q0, where V (q) reaches a local minimum. As with the Morse potential, a separatrix is not always necessary, but we do need a range of closed curves (α) and an action function S(α) to measure their interior C areas. The WKB approximation is an attempt to build a wavefunction around each curve (α), an attempt that fails many times over, but not completely. The idea is to introduce a C complex-valued wave amplitude ψ, which usually obeys Schr¨odinger’sequation,

2 ~ 2 ∂ ψn(q) + V (q) En ψn(q) = 0, −2m q −

in the position basis withp ˆ i~∂q andq ˆ q. Choice of basis breaks the phase plane’s → − → inherent symmetry, but we continue to seek a solution along closed contours of a surface

2 (α) = (p, q) C : α = 2H(p, q) , embedded in the complexiﬁed phase plane. Typically S ∈ we look in a restricted subspace where I(q) = 0, but this extra constraint can be relaxed,

iSn(q)/~ as it is in the center of Fig. 4.8. Making an Ansatz ψn(q) = e , expanding Sn(q) in

powers of ~, and solving on the coeﬃcients, we follow Ref. [52] Ch. 12 to an approximate

140 Black: exact φ 7 e | i I(p) C Gray: WKB ψe7 | i 0 Red: ﬁxed ψe7 | i iπ 1 (e ) 2 phase 0 ψe φ7 shift h 7| i R(p) > 0.999

C R(q)

S◦ Fig. 4.8.: Fixing a WKB use case. Wafefunctions (left), complex domain (center / right).

solution,

q Z 1 i R q 0 0 0 0 ~ ± pn(q )dq Sn(q) pn(q )dq log pn(q) = ψn(q) p e ~ , ≈ ± − 2i ⇒ ≈ pn(q) correct to ﬁrst-order in ~. Critical failure is already guaranteed, because any q axis intersects each (α) at a pair of extremal turning points where p = 0. A zero value causes the C wavefunction amplitude to diverge to inﬁnity. Another failure happens if α is not chosen

R q 0 0 correctly. Then the action integral S(α, q) = pn(q )dq oscillates incoherently around multiple cycles, and the wavefunction averages out to zero by destructive interference. Any closed curve (α) is double-valued over the q axis, but parity symmetry inherent to C 1 2 the kinetic term 2m p makes phase assignment slightly easier. Choosing opposite signs in the exponent, we obtain a real-valued function,

Z q 1 1 0 0 ψen(q) p cos φ0 + pn(q )dq . ≈ pn(q) ~ by summing the left-moving (p < 0) and right-moving (p > 0) parts. From the argument, we H naively expect phase coherence when p dq = (2πn)~ = nh over one complete cycle of (α); C however, this expectation does not account for change of sign at the turning point, where p(q) p(q). Compare Figures 12.1 and 12.2 of [52] with the right of our Fig. 4.8. Instead → − 141 of taking a trajectory through the complex q plane, we hold q approximately constant and allow rotation through the complex p plane23. Either procedure yields the same result, an

iπ/2 extra phase i = e multiplied to the wavefunction ψn(q). Working backwards through

π 1 the approximation, we ﬁnd that ∆S = ~ log(+) log( ) = ~ = h. After two − 2i − − 2 4 H 1 consecutive turning points, the quantization condition reads p dq = (n + 2 )h, with total shift δ = 1 h. Identifying S(α) = H p dq as the area enclosed by curve (α), the quantization 2 C −1 1 −1 condition determines allowable energies αn = S (n + 2 )h , where α = S S(α) . Between the turning points, the WKB wavefunction oscillates similarly to a sinusoid. As

quantum number n increases, ψen(q) shows more nodes, or, crossings of the q axis. Outside the classically allowed region, purely imaginary momentum turns the sinusoid hyperbolic,

and the wavefunction ψen(q) dampens exponentially to zero. The simplest test case is the

24 2 2 harmonic oscillator , α = 2H◦ = ω(p + q ) with action S(α) = (π/ω)α. The quantization ˆ ˆ condition determines that αn = (2n + 1)~ω, exactly the eigenvalues of 2H0 = ~ωN. As mentioned earlier, exact agreement also occurs with the Morse potential, but these two

cases are accidentals. Even for simple harmonic oscillation, the WKB wavefunction ψen(q) is far from perfect, as can be seen in the left of Fig. 4.8. To ﬁx the divergences at the turning points, we can choose a diﬀerent integration contour and avoid them altogether. For

example, integration along the red curve of ◦ in Fig. 4.8 produces a smoothed wavefuntion S 0 ψe7 with seven nodes. The ﬁxed wavefunction then agrees with exact wavefunction φ7 to 99.9% overlap accuracy. Finding the WKB energy levels does not require integration of the corresponding wavefunctions. We now leave this extra step behind, and return to analyzing quartic and sextic double-wells. Up to an Abel-Wick rotation, the quartic and sextic Hamiltonians of Fig. 4.9 are two of the canonical Ramanujan geometries, prepared in Table V of [60]. Their

0 period functions satisfy hypergometric annihilators, b4 and b in the previous article’s A A6

23Abel-Wick rotation p ip produces e(α) from (α). Locally around a turning point, the tangent plane → C C of is a complex p plane, crossed by both (α) and e(α), as in the right of Fig. 4.8. 24ForS deﬁnitions regarding the classical harmonicC oscillator,C refer back to [58] Sections 3 and 4. 142 α 1 = 2H(p, q) = p2 q2 + 1 q4 α 1 = 2H(p, q) = p2 q2 + 4 q6 − − 4 − − 27 ω ω S π S 1 π 1

1 1 2 2

0 α 0 α 1 1 0 2 1 2 10

1 Fig. 4.9.: Eigenvalues (~ = , red), action functions (blue), and residuals 1000 (black). 50 × notation. Around α = 0, standard solutions are written as,

1 3 1 5 2π 4 , 4 6 , 6 2 T4(α) = 2F1 α , and T6(α) = π 2F1 2α α , √2 1 1 −

with harmonic frequency scale factors ω4 = √2 and ω6 = 2 determined at the circular points (p, q) = (0, √2) and (p, q) = (0, p3/2). Abel-Wick rotation requires a covariant change in

0 energy, α 1 α, which leaves bs invariant, but changes b . This explains why period → − A As 2 2 T6 ends up with a relatively odd-looking argument 2α α = 1 (1 α) , see also OEIS − − − A303790 and links therein. The relevant action functions are obtained by deﬁnite integration,

1 R α 0 0 S(α) = 2 0 T (α )dα , and plotted in Fig. 4.9, which interprets the coherent phase condition from a diﬀerent perspective.

1 1 Instead of root solving condition S(α) = (n + 2 )h to ﬁnd αn, we choose ~ = 50 in the ˆ semiclassical limit ~ 0, calculate eigenvalues En of the appropriate H, and plot the points → 1 1 ˆ ( 2 En, (n + 2 )h) next to function S(α). The eigenvalues of either double well H split by 1 + − ± parity, so we take the average En = 2 (En + En ), with En associated to eigenstates of deﬁnite parity ψ± . Averaged eigenvalues do not line up exactly with S(α), but achieve | n i 99.9% accuracy. Actually, the discrepancy is not with the eigenvalues, but with truncation

of S(α) to ﬁrst order in the expansion parameter ~. In the semiclassical limit ~ 0, matrix → eigenvalues pack more densely into the domain α [0, 1], but they continue to disagree with ∈ approximate S(α). A procedure for expanding an exact interpolation S(α) is not well known. 143 The addition of higher-order corrections appears to be some sort of black magic (Compare [24] Table 3.1). Can we ﬁnd one-to-one maps Hˆ S(α) and/or S(α) Hˆ , which become → → exact in the limit ~ 0? The depth of this attractive problem exceeds our current scope, → so the answer will have to remain as another loose end, yet to be tied.

− + A formula for the splitting energies ∆αn = α α is fairly difficult to derive, and not n − n exactly easy to guess. The energy cost from one well to another equals ∆αn, and occurs ˆ off diagonal in the 2 2 splitting matrix, Hn = αnI2 + ~ωn(I2 ρˆ). The first term I2 × − is an identity matrix, which acts on basis vectors ψL and ψR , wavefunctions existing | i | i individually in the left and right wells. Perturbing matrixρ ˆ is not diagonal in this basis. It accounts for tunneling between the two wells. Along a path between mirror-image positions qL and qR, as in Fig. 4.10, the WKB wavefunction amplitude decreases according to,

Z qR −Se(α)/(2~) ψ(qR) = ψ(qL)e where Se(α) = 2i p dq and p(qL) = p(qR) = 0. − qL 6

0 0 Perturbed wavefunctions ψ = ψL + ψR and ψ = ψR + ψL incorporate the | Li | i | i | Ri | i | i possibility of tunneling while remaining normalized when = e−Se(α)/(2~) is small enough.

0 0 Instead of ψL ψR = 0, after perturbing, we have that ψ ψ = 2. By design, this h | i h L| Ri transition amplitude also appears as a coeﬃcient in the Hamiltonian’s transposing term,

0 0 0 0 2 I2 ρˆ = 2 ψL ψR + ψR ψL whereρ ˆ = ψL ψL + ψR ψR and 0. − | ih | | ih | | ih | | ih | ≈

−Se(α)/(2~) Finally we can identify that ∆αn = 8~ωn e . This ”derivation” does not inspire ∝ much conﬁdence, but roughly agrees with the result that ∆α = 4~ e−Se(αn)/(2~). We accept n T (αn) either formula, and refer readers to another proof in [52] Section 12.6. Reﬂecting on the previous developments of [58, 60], we can now begin to appreciate Niels Abel’s prescient research into complex time and double periodicity. The left and center panels of Fig. 4.10 bear a strong analogy to Figure 13 of [58]. Per Harold Edwards’s original analysis, either quartic model,

2 2 1 4 2 2 1 4 2H4(p, q) = p + q q or 2He4(p, q) = 1 + p q + q , − 4 − 4 144 2 Ribbon sections of the torus Uniform domain t C/Z A ﬂattened toric section 2 ∈ 4 = (p, q) C : α = 2He4(p, q) S { ∈ } I(t) R(p) R(p) R q q C L R

1 Se R(q) 2 SS I(p)

tunneling L R(q) C L e R C C C

R p e I p ( ) C R(t) ( )

Fig. 4.10.: Representations of the complexiﬁed quartic tunneling model, α = 1/2.

admits a doubly-periodic time parameter t, and the exact solution p(t), q(t) can be written √ 2 h 1 , 3 i. h 1 , 3 i in terms of Edwards’s ψ(t), with half-period ratio τ = i 2F1 4 4 1 α 2F1 4 4 α (as 2 1 − 1 mentioned in Sec. V of [60]). The exact solution allows us to expand hyperbolic sections of

25 surface 4 around the orthogonal contours , S

n 2 o n 2 o (α) = (p, q) R : α = 2He(p, q) and e(α) = (p, q) R : α = 2H(p, q) . C ∈ C ∈

Now in semiclassical mechanics, the bigger challenge is to integrate a WKB wavefunction on a contour through the complex time plane. Again, the standard contour with I(q) = 0 encounters a divergence at turning points where (α) and e(α) intersect. Can we ﬁx the C C divergence by ﬁnding a path along locally-hyperbolic sections of 4 while avoiding turning S points? The answer is probably yes, but not presently. For expediency’s sake, we reiterate that the task of calculating energy values and their splittings only depends on interior areas such as shown in the rightmost panel of Fig. 4.10. According to the covariance of Abel-Wick rotation, tunneling can be described by as a motion through complex time, along a classically-forbidden trajectory e(α). In terms of probability, tunneling amplitudes are C suppressed exponentially by Se(α), the area beneath e(α). Abel died a century before the C 25Don’t get confused by notation here. The tildes are reversed because the tunneling model puts the double-well on the real p axis. Abel-Wick rotation also requires a transposition τ 0 = 1/τ. − 145 Quartic double-well Sextic double-well 0 0 log(∆α) log(∆α) − −

50 50

α α 100 100 10 10

1 Fig. 4.11.: Tunnel splitting ∆α on a logarithmic scale: eigenvalue diﬀerences (~ = 50 , red), semiclassical estimate (green), and residuals 100 (black). ×

WKB approximation became useful to quantum mechanics. He could not have guessed this beautiful outgrowth, but he may have believed in inevitable consummation anyways.

1 R 1−α 0 0 In both the quartic and sextic examples, complex action functions Se(α) = 2 0 Te(α )dα are obtained by integrating complex periods,

1 3 1 5 4 , 4 6 , 6 2 Te4(α) = 2π 2F1 α and Te6(α) = 2π 2F1 α , 1 1

0 which satisfy b4 Te4(α) = 0 and b Te6(α) = 0 respectively. In both cases, tunneling curves A ◦ A6 ◦ e(α) have a harmonic limit T (0) = 2π centered at (p, q) = (0, 0). Once all real and complex C periods are known and computable, we can evaluate the semiclassical splittings,

4 −1 R 1−αn 0 0 4 −1 R 1−αn 0 0 ~ Te4(α )dα ~ Te6(α )dα ∆αn e 4~ 0 and ∆αn e 4~ 0 . ≈ T4(αn) ≈ T6(αn)

In a complete semiclassical calculation, we start by root-solving S(αn)/π ~(2n + 1) = 0 − for all quantum values αn falling beneath the barrier height α = 1. The set of αn in turn

determines a set of ∆αn. For both double-wells, quartic and sextic, the set of ∆αn agrees with matrix eigenvalue diﬀerences to within 1% (on a logarithmic scale). The combined accuracy of Fig. 4.9 and 4.11 is promising, especially toward the middle of the allowable energy region.

As long as ~ is small, bound states pack densely, and we expect semiclassical estimates 146 to closely match eigenvalues of exact matrices. The question of whether or not to use semiclassical methods can then be argued solely in terms of eﬃciency. Whenever possible, we desire to avoid matrix diagonalization, which scales poorly with number states. On the semiclassical side, root solving and function evaluation are low-complexity tasks, which we can easily loop over quantum index n. Due to sparse density of states, Ammonia is a worst-case example where we could expect inaccurate results, but let’s just try to luck out on the sextic anyways. The constraint from data, (α1 α0)/∆α0 = 1202, determines − a best ﬁt ~ 0.17405, close to the earlier value ~ 0.1751, but still much larger than ≈ ≈ −1 1/50. Nevertheless, the extracted barrier height νB ν01/(α1 α0) 1768 cm matches ≈ − ≈ the expectation value 1769 cm−1 to astounding accuracy! In this case, even though ~ is relatively large, we don’t need to diagonalize a matrix. Recall that vibrational levels occupy a middle ground of the Born-Oppenheimer hierarchy. Rotational levels are one step less energetic, thus they are easier to populate and to observe.

According to thermodynamics, we will often ﬁnd high-J conditions where states pack densely onto the phase sphere. Then we can expect semiclassical methods to work better for rotations than for vibrations. More theory needs developing.

4.5. Rigid Rotors Redux

A simple starting place for the theory of semiclassical rotations is found by considering a quantum ﬁdget spinner. For a laugh, okay, here’s another joke—in Willam Harter’s language, the quantum ﬁdget spinner is also called the Bohr Ring, and admittedly, it is kind of boring 26. Puns and pejoratives aside, let’s not deny the idea of a very small system with one rotational degree of freedom φ, and a classical Hamiltonian

2 α = 2H(Jz¯) = Jz¯ /Iz¯.

ˆ ˆ ˆ ˆ The quantum assignment Jz¯ i~∂φ and φ φ allows a commutation relation [φ, Jz¯] = i~, → − → ˆ ˆ ˆ ˆ which tells us that we can identify (ˆp, qˆ) = (Jz¯, φ). The quantized Hamiltonian 2H(Jz¯) = 26See Ch. 13 of: https://modphys.hosted.uark.edu/markup/GTQM LecturesDetail 2017.html. 147 (a, b, c) = (2, 2, 1) j = 5

Phase z¯ Sphere y¯ x¯ RES

Fig. 4.12.: Symmetric Top Quantization.

ˆ2 1 ±imφ J /Iz¯ has eigenfunctions φ m, = √ e with doubly-degenerate energy levels Em = z¯ h | ±i 2π 2 2 ~ m 27 1 H . This is exactly the WKB result , using Jz¯dφ = 2mπ. As is well known, the lack 2Iz¯ ~ of turning points along the φ coordinate explains why the quantization condition isn’t oﬀset

1 by an extra phase 2 h. This quantization condition applies to any rotational Hamiltonian. Instead of assuming the quantum ﬁdget spinner’s rotational axis ﬁxed, we can allow it to rotate as an oblate symmetric top (c < a = b), with α = H(J) = a(J 2 + J 2 + J 2) + (c a)J 2. x¯ y¯ z¯ − z¯ Euler’s equations of motion determine time dependence, r p p a α ∂tJx¯ = 2 (a c)(a α)Jy¯, ∂tJy¯ = 2 (a c)(a α)Jx¯,Jz¯ = − , ∓ − − ± − − ± a c − such that T (α) = π/p(a c)(a α). As before, the period measures time of one precession − − cycle along a curve (α) with α [c, a]. A few of the curves (α) are drawn in Fig. 4.12, C ∈ C 28 H where they can be seen to bound solid angles S(α) = 2π Jz¯dφ. Action function S(α) − relates to the precessional period by T (α) = ∂αS(α). The quantization condition can be written as either,

1 I m Jz¯dφ = 2mπ or S(α) = 2π 1 p | | , ~ − j(j + 1)

27See Ch. 7, S. 12 of: http://bohr.physics.berkeley.edu/classes/221/notes/wkb.pdf. 28 Solid angle is equivalent to surface area on a unit sphere, dS = d sin(θ) dφ = dJz¯dφ. 148 p with units ~ = 1/ (j(j + 1) chosen to account for normalization of the phase sphere to unit radius. Solving for Jz¯ and substituting it back into H(J), we obtain semiclassical quantum

m2 values αj,m = a + (c a) , where index m ranges from j to j in whole integer steps. − j(j+1) − Checking against matrix calculation, we ﬁnd agreement Ej,m = j(j + 1)αj,m. Thus tunneling must not occur between upper and lower branches of curve (α), as determined by holding C 2 2 J = m / j(j + 1) = constant. Consider that 2∂αS(α) = (2π S(α))/√a α has only z¯ ± − − one solution, S(α) = 2π 1 Jz¯(α) . Since there is no complex solution, we may infer that a − closed curve e(α) does not exist. To encounter tunneling, we need a geometry with a period C annihilator b, which admits at least two linearly-independent solutions. A 2 2 2 In Section 2, we already encountered the rigid rotor Hamiltonian HE(J) = aJx¯ +bJy¯ +cJz¯ ,

and proved its precessional period functions using the annihilator bE. Assuming a < b < c, A real curves (α) chart two separate double-wells, one concave when α [a, b) and another C ∈ convex when α (b, c]. Tunneling happens on complex curves e(α), which rise above the ∈ C separatrix at α = b, and encircle the hyperbolic points at J = (0, 1, 0). A suﬃcient basis ± of phase curves can be written as,

n 3 o x¯(α) = J R : 1 = J J, α = HE(J), α [a, b) , C ∈ · ∈ n 3 z¯ o ex¯(α) = J R : 1 = J ηz¯ J, α = HeE(J), α [a, b) , C ∈ · · ∈ n 3 o z¯(α) = J R : 1 = J J, α = HE(J), α (b, c] , C ∈ · ∈ n 3 x¯ o ez¯(α) = J R : 1 = J ηx¯ J, α = HeE(J), α (b, c] , C ∈ · · ∈

u¯ where Ju¯ iJu¯ takes HE(J) He (J) and J J J ηu¯ J for eitheru ¯ =x ¯ oru ¯ =z ¯. → → E · → · · Complex transformation exchanges either x¯(α) ex¯(α) or z¯(α) ez¯(α), so functionally, C → C C → C either transformation can be seen as the geometric half of an Abel-Wick rotation. The

3 surface J R : 1 = J ηu¯ J is a circular hyperboloid, which completely intersects the { ∈ · · } phase sphere in theu ¯y¯ plane. Along the unit circle of intersection, upper and lower branches

of a real curve (α) meet complex curve e(α) at exactly four points. Away from these points, C C closed curves e(α) loop between opposing branches of (α), so we can expect to ﬁnd tunnel C C 149 (a, b, c) = (1, 4, 5) j = 6

RES

z¯ y¯ x¯ Phase Sphere

Fig. 4.13.: Tunneling Geometries.

ˆ ˆ splitting of Jz¯ doublets (or in quantum-speak [HE, Jz¯] = 0). 6 Before we get too far into analysis of the quantum asymmetric top, let’s pause brieﬂy to look back on the quantum symmetric top. William Harter says (by private communication) that the absence of tunnel splitting is entirely explained by symmetry. This is a true assertion, ˆ for Jz¯ is a valid generator for the Lie group U(2) of circular symmetry. Furthermore, the ˆ2 ˆ commutation relation [J , Jz¯] = 0 implies m a good quantum number, and time reversal ˆ ˆ2 requires H a function of Jz¯ . However, this is a fully quantum explanation, one which does not make use of semiclassical geometry. If we try to apply a transformation Ju¯ iJu¯ we → can chooseu ¯ =x ¯ oru ¯ =y ¯ but notu ¯ =z ¯ because Jz¯ must remain real. Meanwhile, the

3 |m| intersection of a hyperboloid J R : 1 = J ηu¯ J with either plane at Jz¯ = { ∈ · · } ±√j(j+1) results in level hyperbolae e(α), which do not connect between upper and lower branches of C curve (α). In terms of the symmetric top’s semiclassical topology, the complexiﬁcation of C each curve (α) is a Reimannian surface, which decomposes to two unconnected copies of the C harmonic hyperboloid ◦. As we have already mentioned, the situation with the asymmetric S top is entirely diﬀerent. Figure 4.13 clearly shows that real and complex curves link together to form a multiply-connected toric section, which does allow tunneling along green curves

150 eu¯(α) and between opposing branches of blue curves u¯(α). Now we return to the task of C C taking integrals along these curves. To further develop the vibrational-rotational analogy, we start with action function,

I p 2 2 z¯ α a cos(φ) b sin(φ) S a, b, c α = − − dφ, E pc a cos(φ)2 b sin(φ)2 − − which measures the solid angle subtended between curve z¯(α) and the Jz¯ = 0 C equator. Similarly, between x¯(α) and the Jx¯ = 0 equator the solid angle is C x¯ z¯ S a, b, c α = S c, b, a α . Neighboring intersection points of u¯(α) eu¯(α) are E E C ∩ C located by extremal angles in theu ¯x¯ plane,

√α b √b α either γ± = arcsin − or γ± = arcsin − , ± √c b ± √b a − − when α (b, c] or α [a, b) respectively. The tunneling action integral, ∈ ∈ Z γ+ p 2 2 z¯ α b cos(γ) c sin(γ) S a, b, c α = 2i − − dγ, eE p 2 2 ± γ a b cos(γ) c sin(γ) − − − determines the solid angle subtended by curve ez¯(α), when the leading sign is chosen to C z¯ make Se (α) positive and real-valued. To calculate the solid angle of ex¯(α), again we E C permute arguments and obtain Sex¯ (a, b, c α) = Sez¯ (c, b, a α). Essentially, this is the standard E | E | presentation, as was originally reported by Harter in the Springer Handbook of Atomic, Molecular and Optical Physics [50]. The semiclassical formulae,

4 1 u¯ 1 u¯ ~ − 2 SeE (αj,m)/~ SE(αj,m) = m~, ∆αj,m = u¯ e , ~ = p , TE(αj,m) j(j + 1)

29 determine levels αj,m and their tunnel splittings ∆αj,m. To obtain a complete list of levels, the formulae must be solved in both high-energy and low-energy double-wells. The naive solution procedure calls for a considerable amount of geometric integration. A more reﬁned

approach takes advantage of the annihilator bE. A u¯ u¯ The basic idea is to deﬁne functions SE(α) and SeE(α) not as geometric integrals over

angular coordinate, but relative to solutions of annihilator bE. In section 2, we already A 29The splitting formula carries over from the previous section with a hidden factor of 2 to account for two tunneling attempts per classical cycle. 151 u¯ u¯ found a set of periods T (α) such that bE T (α) = 0. The corresponding action functions E A ◦ E follow from integration along the energy coordinate, Z α−a Z c−α x¯ x¯ 0 0 z¯ z¯ 0 0 SE(α) = 2π TE(a + α )dα and SE(α) = 2π TE(c α )dα , − 0 − 0 − Z b−α Z α−b x¯ y¯ 0 0 z¯ y¯ 0 0 SeE(α) = i TE(b α )dα and SeE(α) = i TE(b + α )dα . 0 − 0 Integrals over dα0 can be taken numerically, but instead, we optimize the process by

u¯ numerically solving bE ∂α S (α) = 0 with three pieces of initial data. The mid-range A ◦ ◦ E energies α = (a + b)/2 and α = (b + c)/2 both determine an integration limit γ± = π/4. ± According to Mathematica, the tunneling integrals are elliptic integrals, a + b √a b a c a c Sex¯ = I(c, b, a) = 2√2 − 2Π 1 − K − , E 2 √b c − c b − c b − − − b + c √c b c a c a Sez¯ = I(a, b, c) = 2√2 − 2Π 1 − K − , E 2 √b a − a b − a b − − − and after a little more integral analysis, we ascertain that, a + b b + c Sx¯ = 2π + iI(c, a, b) and Sz¯ = 2π + iI(a, c, b). E 2 E 2

u¯ u¯ 30 The values of TE(α) and ∂αTE(α) are also known in terms of elliptic integrals , which allows for very rapid calculation of initial data at α = (a + b)/2 and α = (b + c)/2. Then the ordinary diﬀerential equation can be integrated numerically to produce all the necessary functions in computable form. This type of procedure is standard fare in the practice of Ordinary Diﬀerential Equations. In this context, it is a boon, which sets us up to solve energy levels and their splittings without evaluating any more integrals! The vibrational-rotational analogy reaches fruition in Fig. 4.14, the direct successor of earlier Figures 4.9 and 4.11. To show inversive symmetry, we choose (a, b, c) = (1, 2, 5) and (a, b, c) = (1, 4, 5) as in Figures 4.2 and 4.13. The combined transformation,

b b0 = a + c b and α α0 = a + c α, → − → − 30See also https://functions.wolfram.com/EllipticIntegrals/EllipticPi/. According to Christoph Koutschan’s m ∂ m m2 ∂2 m m2 ∂3 HolonomicFunctions, annihilator√ bΠ = 3 6(1 7 ) m 4(2+ 11 ) m 8 (1 ) m uniquely π A2 − − − − − − deﬁnes Π( 1 m) = 4 √2 + (1 2 )m + ... by bΠ Π( 1 m) = 0. If elliptic functions are not readily computable,− initial| data can be− calculated more slowlyA ◦ by− numerical| integration. 152 2π SE (a, b, c) = (1, 2, 5) 2π SE

π π

1 3 4 α 1 32 α

−50 −50

−80 −80

log(∆α) (a, b, c) = (1, 4, 5) log(∆α)

Fig. 4.14.: Asymmetric top quantization at j = 30: eigenvalues (red), eigenvalue splittings (orange), semiclassical estimates (blue, green), and residuals 100 (black). ×

acts as label permutationx ¯ z¯ on all initial data and the functions T u¯(α), while leaving ↔ E annihilator bE invariant. Consequently, semiclassical functions depicted in the left and A right of Fig. 4.14 relate by reﬂection through the central value α = 3. Pair-averaged matrix eigenvalues have the same reﬂection symmetry, because the eigenvalues of

ˆ 0 ˆ ˆ ˆ ˆ ˆ ˆ2 ˆ2 ˆ2 H (J) = (c + a)J J HE(J) = cJ + (a + c b)J + aJ , E · − x − x x equal to those of Hˆ (Jˆ) reflected through (c + a)/2. Residual differences do not change from left to right, but percent errors appear slightly worse with a lower-energy separatrix. For j = 30, levels and tunnel splittings are calculated with better than 99.9% and 97% accuracy, respectively. These accuracy bounds persist as low as j = 10, but small j inaccuracies are not too bad either. The parameter values of Fig. 4.14 are chosen somewhat arbitrarily, so in fig 4.15, we allow for variation of both b and j. On left, when b changes, black points are calculated using the semiclassical method and closely match underlying curves, which are calculated by exact matrix diagonalization (compare with [49] Fig. 31.1). The most important feature to realize is the failure of the approximation near the separatrix energy. Sometimes a level is missing, as is indicated by the red points. If necessary these points can be interpolated.

153 α (a, c) = (1, 5) t(ms) Mathematica AbsoluteTiming at 2.7 GHz 300 t 0.321j2 4 ≈

2 100 t 16.7 + 1.3j ≈ 1 b j 1 2 3 4 5 30210

Fig. 4.15.: Asymmetric top Energy levels for j = 10 (left), algorithm timing test (right).

The right panel shows a timing test where the semiclassical method outperforms the matrix method after j = 10. Data is obtained for (a, c) = (1, 5) with randomized b [1.5, 4.5], and ∈ then averaged over fifty trials. The linear fit places the initial cost of function construction somewhere around 20 milliseconds, while calculation of one level and its splittings costs less than one millisecond. Diagonalization times are fit to a quadratic function, but in the long run, theory gives an expected complexity of (j2+δ) where δ [0, 1]. O ∈ In practice, the accuracy of Fig. 4.14 and the speed of Fig. 4.15 suggest that the semiclassical method could be a competitive technique for extracting the (a, b, c) parameters. This is certainly true above some cutoff, in our timing test j = 10. The semiclassical method can also compete below the linear-quadratic crossing, because function construction is a one-time cost per choice of (a, b, c). To calculate all levels for j = 1, . . . , jmax it takes either t 16.7 + (1.3/2)j2 or t (0.321/3)j3 milliseconds. The constant term of the ≈ max ≈ max semiclassical time is not integrated, and with this time saving, equal efficiency occurs earlier, in our test around jmax 8. For example, water vapor (H2O) would make a good test case for ≈ parameter extraction with levels known up to jmax = 8 and previously reported expectation values (a, b, c) (27.9, 14.5, 9.3) in units cm−1 [40]. One further obstacle is that we can not ≈ expect molecules to always behave like rigid rotors, especially as kinetic energy increases

154 with j. Centrifugal distortion is more than just a bane to data analysts. It is also a way to make the quantum symmetric top vastly more interesting.

4.6. Quantum Symmetric Tops

The rigid-rotor assumption completely contradicts the Born-Oppenhiemer approximation, which allows for vibration and rotation in every electronic bound state. Even in one ﬁxed electronic state, a ball-and-spring picture, such as Fig. 4.16, is more accurate than an overly-simplistic ball-and-stick picture. To fully account for stretching and bending, theory must admit the possibility of rovibrational interactions such as Coriolis coupling, and the possibility of pure-rotational perturbations due to centrifugal distortion. However, for the present article, we will continue with a strict and simple interpretation of Born-Oppenheimer, by keeping rotations and vibrations separate. Even then, it is inevitable that high-energy rotations will introduce centrifugal forces, in turn causing changes in molecular shape. What can we do?

Centrifugal force is proportional to J2, so the condition of high-J should be expected to exaggerate its eﬀect. When examining spectral data, energy shifts are seen as translations of spectral lines. According to the vibrational-rotational analogy, we already know that the theoretical procedure of adding higher-order terms to the Hamiltonian can account for such shifts, so an eﬀective solution is straightforward. Instead of HE(J), we consider a

2 P i j k perturbative form, H(J) = α0J +HE(J)+ i,j,k ci,j,kJx¯Jy¯Jz¯ with i+j +k > 2. A maximally permissive scheme still requires i + j + k = even to preserve time-reversal symmetry. Up to fourth order, this allows for 15 undetermined coeﬃcients ci,j,k. As has always been the case in molecular physics, imposition of symmetry makes analysis more tractable. Instead of working out the fully general problem, we will continue with a few more starter examples, as depicted in Fig. 4.17 and as listed in the left column of Table 4.3. These polynomial functions have a very rich history31, but they are not yet entirely well-understood.

31Edouard´ Goursat’s Etude´ des surfaces. . . http://www.numdam.org/item/ASENS 1887 3 4 159 0/ and 155 z¯ y¯ x¯

Fig. 4.16.: Ball-and-spring XY6.

More than two thousand years ago the Greeks discovered the platonic solids, and these few beautiful shapes are still being studied today. By transformation analysis, we can readily prove that any molecule with tetrahedral, octahedral or icosahedral symmetry must have

Ix¯ = Iy¯ = Iz¯, i.e. must behave like a spherical top at low-J. Any of the three perturbations

0 HX (J) with X = T,O or I, as listed in Table 4.3, contributes to a classical Hamiltonian,

2 0 HX (J) = α0J + HX (J), which eﬀectively describes non-rigid rotations of a molecule with polyhedral symmetry. The tetrahedral perturbation violates time-reversal symmetry, but we

0 include it anyways, if only because it is isoperiodic to canonical Ramanujan geometry H3 of [60], how curious! Perturbing terms account for the possibility of centrifugal distortion by giving hills and valleys to the rotational energy surface. For example, the octahedral molecule Sulfur hexaflouride (SF6) has hard S-F bonds along thex ¯,y ¯ andz ¯ axes. These bonds resist stretching, but bend easily when the molecule rotates around a soft intermediary axis such as (1, 1, 1). The physical idea is that centrifugal distortion lowers rotational energy around soft axes. This is reflected in the effective Hamiltonian RES by occurrence of hills along hard axes, and valleys along soft axes.

ˆ ˆ 0 When quantizing HX (J), the energy levels of HX equal to those of HX oﬀset by a shift ˆ2 ˆ 0 α0j(j + 1) = J , so we only need to ﬁnd energy levels of one chosen H . This is seen most h i X

Felix Klein’s Lectures on the Ikosahedron https://archive.org/details/cu31924059413439. 156 TOI

Fig. 4.17.: Sphere curves with Tetrahedral, Octahedral, and Icosahedral symmetry.

Table 4.3.: Hamiltonians with Tetrahedral, Octahedral, and Icosahedral symmetry. Hamiltonian Perturbing Terms Period Annihilator √ 0 3 2 3 2 3 2 2 2 H = J + (J 3Jx¯J ) (J Jz¯ + J Jz¯) bT = 8 α 9(1 3α )∂α T z¯ 2 x¯ − y¯ − 2 x¯ y¯ A − − = J 0 J 0 J 0 (after a rotation J R J) 9α(1 α2)∂2 x¯ y¯ z¯ → · − − α 0 4 4 4 H = 2J + 2J + 2J bO = 9 (6 5 α) O x¯ y¯ z¯ A − 2 2 2 2 4(Jx¯Jy¯) + 4(Jy¯Jz¯) + 4(Jz¯Jx¯) 16(12 22α + 9α )∂α ' − − +16 (2 3 α)(1 α)(2 α)∂2 − − − α 0 6 2 2 4 2 2 2 2 H = J 5(J + J )J + 5(J + J ) J bI = 5 (5 21 α) I z¯ − x¯ y¯ z¯ x¯ y¯ z¯ A − 4 2 2 4 2 2(J 10J J + 5J )Jx¯Jz¯ +4(5 + 44α 81α )∂α − x¯ − x¯ y¯ y¯ − +4α(5 + 27 α)(1 α)∂2 − α

obviously true from the semiclassical perspective. Level curves of α = HX (J) diﬀer from

0 0 those of α α0 = H (J) only by the energy shift α α = α + α0. Using matrix methods, − X → a nuance is encountered due to the fact that components of the quantum Jˆ-vector do not commute. We will return to this issue shortly. To evaluate the quantization condition, again we need to integrate the real and complex action functions along curves (α) and e(α). C C As with HE(α), we can prove that action-angle coordinates (λ, φ) = (Jz¯, φ) are canonical

0 for all three choices X = T,O or I. Inputing the Hamiltonians HX (λ, φ) to algorithm DihedralToODE, we obtain the annihilators in the right column of Table 4.3. Up to initial data, these annihilators mostly solve the semiclassical quantization problem, but we still

157 Tunneling Tunneling on soft axes on hard axes

z¯ α [ 2 , 1] RES α [1, 2] ∈ 3 y¯ ∈ x¯

Fig. 4.18.: An Octahedral RES and its associated tunneling geometries.

need to calculate initial data. Higher symmetry ﬁrst requires us to revise the procedure for transforming between real and complex contours.

The asymmetric top Hamiltonian HE(J) has three planes of reflection symmetry,x ¯y¯,y ¯z¯, andz ¯x¯. Two of these planes,x ¯y¯ andy ¯z¯, contain the branch points u¯(α) eu¯(α), whenu ¯ =z ¯ C ∩C andu ¯ =x ¯ respectively. With polyhedral symmetry, we utilize a similar complexification strategy, and obtain curves eu¯(α) by Abel-Wick rotation on an axis of symmetry Jv¯ with C v¯ =u ¯. Instead of Cartesian indices (¯x, y,¯ z¯), we make use ofn ¯ = 2¯, 3¯, 4,¯ or 5¯ for an axis Jn¯ 6 with dihedral Dn¯ symmetry. For the octahedral geometry depicted in Fig. 4.18, the soft-axis curves 3¯(α) connect to their tunneling counterparts e3¯(α) in equatorial planes orthogonal to C C ¯ any 2 axis, while hard-axis intersections 4¯(α) e4¯(α) fall into equatorial planes orthogonal C ∩ C to any 4¯ axis. The 4¯ axes can be chosen as (¯x, y,¯ z¯). Branches of ¯(α) encirclingz ¯ connect C4 to nearest neighbors on axesx ¯ ory ¯ by curves e4¯(α) after transforming either Jy¯ iJy¯ C → or Jx¯ iJx¯ respectively. Similarly, nearest-neighbor branches of curves ¯(α) connect via → C3 ¯ e3¯(α) after transforming J2¯ iJ2¯ on any of the six 2 axes. The icosahedral case is slightly C → more complicated due to higher symmetry, but essentially the same. Once curves n¯ and C en¯ are defined over their respective energy ranges, any technique can be used to calculate C 158 S Octahedral, j = 100 S Icosahedral, j = 200 O 2π I

3 2 π 0

20 −

40 − log(∆α) α log(∆α) α 2 1 2 5 0 1 3 − 27 Fig. 4.19.: Octahedral and Icosahedral quantization, same conventions as Fig. 4.14.

Table 4.4.: Initial data for Octahedral, and Icosahedral action functions.

0 00 0 00 X α SX (α) SX (α) SX (α) Se(α) SeX (α) SeX (α) O 5 6.063688 1.495838 2.836477 0.196938 1.27896 1.625205 6 − − − 3 5.816795 1.133863 1.122413 0.516411 1.011125 0.107365 2 − I 5 6.210605 0.882921 2.827018 0.062569 0.738854 1.854874 − 54 − − − 1 6.044973 0.593385 0.669005 0.269061 0.503328 0.004684 2 − the initial data. To obtain the values of Table 4.4, we use a mixture of series expansion and numerical integration. From these values, all relevant action and period functions can be obtained by numerical integration of the ordinary differential equation AbX (∂α S(α)) = 0. ◦ ◦ In practice it is much more difficult and time consuming to calculate the necessary matrix egeinvalues, and there are many chances to make a mistake. As mentioned above, the first problem is non-commutation. Following Harter and Patterson’s original approach in [44, 45], we sidestep indeterminacy issues by using tensorial operators instead of applying the naive map from J ~Jˆ. There is only one Octahedral-symmetric tensor at fourth order, but → we can translate the energy scale by adding in a term like Jˆ4. To locate splitting around ˆ ˆ HO = 1, while limiting the range of energies HO [2/3, 2], we adapt, h i h i ∈ 8 r 5 6 Hˆ = Tˆ4 + Tˆ4 + Tˆ4 + Jˆ4, O 10 0 14 4 −4 5

159 ˆk from the earlier form. Tensorial terms Tq have matrix elements,

1 2 j ˆk j k,j,j 1 (2j + k + 1)! k m Tq n = j k j Cq,m,n with j k j = ~ . h | | i h || || i h || || i 2k (2j k)!(2j + 1) − The Clebsch-Gordan coefficients Ck,j,j multiply a reduced matrix element j k j , as q,m,n h || || i prescribed in [47]. Asymptotically, the reduced matrix element tends to one, i.e. j → ∞ ˆ ˆ4 ˆ4 ˆ4 causes j k j 1. In the same limit, HO 2(J + J + J ), so this naive form can be h || || i → → x y z used as a substitute if necessary. In the Icosahedral case, no simple approximation in terms of Jˆ exists, so there is no choice but to use the dreaded Clebsch-Gordan coefficients32. ˆ ˆ If the matrix representations of HO and HI are correctly defined, we expect that high symmetry will cause multiple degeneracy in the corresponding spectrum of matrix eigenvalues. With the asymmetric top, we only found double wells onx ¯ andz ¯ axes. The ˆ ˆ Octahedral Hamiltonian HO has four 3¯ axes and three 4¯ axes. Similarly Icosahedral HI has ten 3¯ axes and six 5¯ axes. Instead of doublets, we expect to find multiplets, with degeneracy 2 n¯ , where n¯ is the count of similar axes. Again, due to tunneling between branches | | | | of (α), mean energy values split according to the eigenvalues of a traceless perturbation C matrix Hˆ , but this time the dimension of Hˆ must be d d where d = 2 n¯ rather than 2 2. × | | × Naively we could expect that Hˆ is just an adjacency matrix describing the branch topology

of the corresponding curves u¯(α); however, this hypothesis is correct only sometimes. C A most remarkable accomplishment of Harter’s symmetry theory is its usage of the Frobenius reciprocity theorem to describe tunneling resonance on cyclic symmetry axes. The

ˆ imφ basic idea is that the Jz¯ eigenfunction φ m e gets into tune with local cyclic Cn h | i ≈ ˆ ˆ symmetry whenever Jz¯ = Jn¯. In the list of energy levels, this symmetry resonance is seen as a sequence of multiplets, which repeats modulo n. In theory, correlation tables33 help to explain which irreducible representations the multiplet eigenfunctions ultimately belong to; however, nomograms such as found in [44] are also necessary to account for oﬀset from the m = 0 groundstate. Let’s take a deﬁnite example and see what happens. After diagonalizing

32Apologies for this joke. Check SPSD, or just use Mathematica ClebschGordan. 33Refer to SPSD Ch. 5, around p. 385. For another approach, see also [57] (due for an update), and https://demonstrations.wolfram.com/MolienSeriesForAFewDoubleGroups/. 160 ˆ HO at j = 100, we read the eigenvalues toward the separatrix energy and find the following patterns in their partitioned first differences,

10010 00100 01001 00100 10010 00100 01001 00100 4ˆ : ... AT E TT ETA TT AT E TT ETA TT

0010100 1001001 0010100 1001001 0010100 1001001 3ˆ : ... TET AT T A TET AT T A TET AT T A

In this notation, a 1 means split, 0 means stay, while A is a singlet, E a doublet, and T a triplet. According to theory, the groundstate multiplets are A2T2T1A1 and A1T1E, so we obtain the correct offsets according to 100 mod 4 = 0 and 100 mod 3 = 1. Now we describe a creative way to discover the splitting matrices along the hard 4¯ axes. Start by assuming that Hˆ m is a signed adjacency matrix, with either 1 for every element O ± between nearest-neighbors, and all other elements equal zero. Assuming transpose symmetry, this amounts to only 212 = 4096 possible matrices. After enumerating these matrices and diagonalizing them, we inspect eigenvectors34 to find a complete set of signed adjacency matrices, as written out and depicted in Fig. 4.20. Wait a second, something amazing is happening! What’s going on here? The usual case A1T1E is just a plain adjacency matrix. The two odd cases T1T2 and ET2A2 share a common feature. The product of signs along any graph 4-cycle equals 1, while the product of signs along any 3-cycle equals 1. − Physically, this can be interpreted as symmetry resonance, which allows tunneling around any 4¯ axis, while disallowing tunneling around any 3¯ axis via destructive interference. We could hope that such a feature would persist in other similar calculations, but unfortunately generalization is not straightforward. Checking by brute force combinatorics, we find that the

1|2 splitting matrix HO for C3 multiplet T1ET2 can not be written as a signed adjacency matrix. Icosahedral splitting matrices have higher dimensions, involve a wider range of phases, and have non-integer splittings. In these remaining cases (for the time being), Harter’s theory gives the only acceptable explanation. In practice, comparison of tunnel splitting values is irrespective of the labeling of

34The eigenvectors have well-deﬁned transformation properties, or we can compute overlaps with Harter’s canonical basis vectors. 161 A1T1ET1T2 ET2A2

0 1 1 0 1 1  0 1 1 0 1 −1 0 1 1 0 1 1  1 0 1 1 0 1 1 0 1 −1 0 1 1 0 −1 1 0 −1 ˆ 0 1 1 0 1 1 0 ˆ 1|3 1 1 0 1 −1 0 ˆ 2 1 −1 0 1 −1 0 HO = 0 1 1 0 1 1 HO =  0 −1 1 0 −1 −1 HO = 0 1 1 0 1 1  1 0 1 1 0 1 ± 1 0 −1 −1 0 −1 1 0 −1 1 0 −1 1 1 0 1 1 0 −1 1 0 −1 −1 0 1 −1 0 1 −1 0

Fig. 4.20.: Tunnel splitting adjacency matrices, black = 1, red = 1 (contours for j = 30). −

ˆ n eigenstates and even irrespective of multiplet ordering. Given the matrices HO, we can calculate multiplet splitting by diagonalization. More expediently, we can simply reference level splitting ratios for each particular multiplet from Harter’s original series on Octahedral and Icosahedral symmetry [44, 45, 47]. After identifying patterns in the first differences, we divide splittings by the appropriate ratios, and average over each multiplet to obtain a set of ∆Ej,m, with the expectation that ∆Ej,m ∆αj,m. For both octahedral and icosahedral ≈ cases, semiclassical splitting and tunneling values are compared to matrix eigenvalues in Fig. 4.19. Using an entirely new method to draw this graph is a significant milestone for the theory, but we should also emphasize that more work remains to be done to fully understand how real symmetry analysis transfers into complex domain.

4.7. Conclusion

Next to the lifetime collected works of Landau, it could be argued that the entire dissertation, including this ﬁnal chapter, only deserves an incomplete rating. Undoubtedly, this view will be taken up by many opponents, because it is a popular one. Granted, there are more

162 interesting questions to answer, and more powerful algorithms to write. Given more time and better resources, we would like to continue developing Hamilton-Abel theory. However, in this conclusion we must argue the opposite perspective, that enough has already been done. The minimum, in theory or in practice, is that the writing describes reproducible results, and that the results themselves are original or unexcelled. It is always a tall and momentary order, but let’s just reflect on what we’ve already seen. The circle itself, as it exists algebraically, goes all the way back to antiquity. Many, many years later, Sommerfeld’s work on geometric physics and quantum mechanics could be reproduced and further developed by Landau. Then Landau’s work could be reproduced and extended by Harter. Now we have, at the very least, even more of a geometric language to work with. We have improved integral analysis of all examples displayed in Fig. 12 and listed in Table IV of the dissertation Prelude. No, we did not succeed in applying Creative Telescoping to every possible plane curve, nor to every possible sphere curve. If we had, the earth would not shake by our sole action, and science itself would still have more work to be done. Instead, in the simple pendulum chapter [58] and in this chapter, we perform easily reproducible experiments using a fidget spinner, ha ha ha! Let us again emphasize that the connection between the plane pendulum and the asymmetric top has never been as clear as it is today. If we extract the parameter from period-energy data, or if we extract parameters (a, b, c) from a set of spectral data, we now can realize similarity of results, where before we found mostly ignorance. In the preceding chapter [60] on Ramanujan’s premiere article [77], we developed Creative Telescoping as much as time could allow, but the purpose for doing so was somewhat obscured by a Mystery. As nice as it would be to divert into studying Modular functions, our true purpose, so far, is only to redact the amount of difficult analysis needed while advancing through the vibrational-rotational analogy. It is an accomplishment to obtain relevant annihilators, and then we can move quickly through a range of similiar examples, as we have done with Figures 4.9, 4.11, 4.14, and 4.19. When making these graphs, we develop

163 newly emerging computational procedures in another direction entirely. Semiclassical quantization of Ramanujan geometries was ﬁrst discussed in [8]. Now this chapter seems to be the ﬁrst written report where rotational quantum values have been calculated using a combination of Creative Telescoping and the semiclassical WKB approximation. Following a thorough analysis of the imminently important asymmetric top, we also succeed in integrating canonical examples featuring octahedral and icosahedral symmetry. Again, this is not ”everything”, but it is a good note to end on. At least I think we can agree that there isn’t another Platonic solid with higher symmetry than that of the icosahedron!

Acknowledgements

William Harter contributed many interesting lectures and conversations in support of this work, and helped to ensure that it could be funded to completion. Calculations and drawings are the author’s own work (usually with the aid of Mathematica [54]).

164 5. Coda (on Prajna)

Western science keeps knowledge and wisdom separate, and it is largely an enterprise for knowledge only. Wisdom sometimes falls out of a historical lesson, if the author or speaker bothers to give any history at all. In Asian science, the lines are blurred. A pendulum is just a wheel with one spoke. A wheel with one spoke is harder to turn than a wheel with three spokes, but three spokes are easier to turn than eight or twenty-four. A wheel in the hand is easier to turn than a wheel in the sky. Someday the wheel turning student becomes the wheel turning teacher. The wheel turning teacher needn’t ever become the wheel turning king (or the wheel-breaking idiot), but might on occasion become the wheel turning ﬁsherman1. Some of the wheels from China now turn freer and easier than ever, so we can really get some interesting work done. Thanks again Lao Tzu and Chuang Tzu, for safe passage! In the original Pali cannon [17], the wheel symbolizes Dhamma, a concept that comes down to the Buddhists from the earlier Hindu Purusartha. In a rough, imperfect translation, Dharma, Artha, Kama, and Moksha in the Greek language would rather be Logos, Ethos, Eros, and Pathos. For the scientist the Dharma-Artha combination is everything. The Brahmacharya is rarer and rarer these days, so it is often joked that, to the scientists, Kama and Moksha, whether taken together or separately, are nothing important. They think they have it but they don’t, or they think they don’t have it but they do. Even more diﬃcult to apprehend is the Moksha-Pathos connection. It is sometimes expressed by poets, as in Percy Bysshe Shelly’s ode To a Skylark 2. It is sometimes screamed about by punk rockers, as in the Black Angel’s 2017 smash hit album Death Song. Speaking for the scientists, perhaps Max Planck said it best as Planck’s Sociological Principle (quoted from https://en.wikipedia.org/wiki/Planck’s principle):

A new scientiﬁc truth does not triumph by convincing its opponents and making

1Not parsing? Try: Aldous Huxley, The Perennial Philosophy, Chapter VI [53]. 2Online: https://www.poetryfoundation.org/poems/45146/to-a-skylark. 165 them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it. . . . An important scientiﬁc innovation rarely makes its way by gradually winning over and converting its opponents: it rarely happens that Saul becomes Paul. What does happen is that its opponents gradually die out, and that the growing generation is familiarized with the ideas from the beginning: another instance of the fact that the future lies with the youth.

In other words, fascism itself cannot be educated into non-existence, and Exodus is just a set up for Deuteronomy, where Moses dies before ever reaching the promised land. The theme of untimely death, important to Moksha Shastra, also occurs in the Chinese drama of Liang Shanbo and Zhu Yingtai, and in Chikamatsu Monzaemon’s famous bunraku play, Shinju Ten no Amijima [23]. When you go this way, combining Dharma and Artha to try and attain Anuttara Samyak Sambodhi, or at least what the Japanese call Shin-ri-kyo, the other issue is whether or not you’ve become involved in some sort of conspiracy between religion, politics, and academics. It’s like you’re driving a ﬁretruck or an ambulance through a Mahayana Riot, a Holy War, a Gender War, a Music/Dance War, a Race War, or through Samsara itself—for ten or ﬁfteen consecutive years, with twenty or thirty left to look forward to (if you can survive the dehydration attack). You constantly have to check the rear view mirror to make sure there aren’t any terrorists following behind. What is the next attack going to be, when is it going to happen? Who will it be this time? The Branch Davidians, the Covenant the Sword and the Arm of the Lord, the Trappists, Patriot Front, or the University of Wherever? Zen Master Seung Sahn said ”Only don’t know”, and yes, that is a good teaching. But when you are done with Zazen practice, there are still a lot of issues in this world that need working on, and there is always another horrible lesson to learn. The analogy between Aum Shinrikyo and the Manhattan Project is an instructive one3. Vastly many more Japanese civilians were killed in the Atomic Bombings of Hiroshima and Nagasaki on 6 and 9 August 1945 than were killed in the Sarin gas attack on the

3Compare Richard Rhodes Making of the Atomic Bomb [78] with Huraki Murakami Underground [69]. 166 Fig. 5.1.: A Ch¨od dancer. Taken as Fair Use from Julian Stephenson, Volume 13: Propogating the Lotus Sutra, p. 10. (Fair use is a form of propagation). Compare with: https://www.metmuseum.org/art/collection/search/69054.

167 Tokyo subway station 20 March 1995. Julius Robert Oppenheimer (1904-1967) was no Shoko Asahara, but he certainly sounded like a cult leader when making this famous pronouncement, a misquotation of Bhagavad Gita, ”I am become death, the destroyer of worlds”4. Ayurveda agrees with the Natya Shastra5 that the phrase ”Kala / asmi” can be more accurately translated as ”Time / I am”, and this reading is consistent with more professional translations available online6. It is important to realize that there is an entire spectrum between Aum Shinrikyo and the Manhattan Project. You don’t want your business to fall anywhere on that spectrum! There is an entire spectrum between Richard Feynman and Kenichi Hirose. You don’t want your life to fall anywhere on that spectrum! It so happened that while I was finishing this dissertation not one, but two aging, mendicant women solicited me for food, one after the other. Okay, we are guilty of some inappropriate socialization and meat eating (that is Murder Karma). Having suffered like that, we are feeling better now, maybe. These were two very difficult cases, because we have all been taken advantage of, trampled upon, and hatefully abused; then put together to suffer together, again and again. That is cyclic nature, and close to my understanding of Samsara. The second time this happened, I couldn’t very well understand anything the poor woman said about her theory on Bermuda Triangle Math or how the Rinpoches were involved with BDSM and verbal abuse. When she said that she wasn’t speaking in tongues and didn’t need any medication, sadly, that sentiment made a lot of sense to me. So I decided to try and help her, and gave her as much food as she would accept. This mad woman dancing her part of a makeshift Chödshowing great Moksha in the street and in an empty barroom, at the edge of the pandemic, it struck me like a Buddhaverse lightning bolt. This is what I found. The goddess of the Vimalakirti Sutra [98], Chapter 7 is likely an emanation of Avalokiteshvara. Much, Much later in time, ”many kalpas”, this Avalokiteshvara shatters into Dalit form according to her (or his?) inability to do

4You can watch him say it on Youtube: https://www.youtube.com/watch?v=lb13ynu3Iac. 5Online: https://www.wisdomlib.org/hinduism/book/the-natyashastra/d/doc209700.html. 6For more sources, see: https://www.wisdomlib.org/deﬁnition/kala, and for a translation of the original source see: https://www.holy-bhagavad-gita.org/chapter/11/verse/32. 168 anything about the other problem. The insects are going homeless due to deforestation and Crazy Real Estate, while ”Twittler”7 rants online. According to Lama Lodru Rinpoche8, Amitabha then remakes Dalit Avalokiteshvara in a new form as Mahakala/Mahakali (and we believe that dung beetle magic may be involved in this process). I know this sounds crazy. Please realize the many diﬀerent forms of Shiva (including Rudra, Shmashanavasin, and Mahakala/Mahakali) make any such story mysterious and dangerous. Many students are too curious about Mahakala/Mahakali, and it really is better not to ask. They are Buddhist dharmapalas, and appear in the Natya Shastra in association with Bibhatsa Rasa9. The name could mean ”Great Time”, but they are not exactly having a great time. Their devotees sometimes dance a Cham for hours and hours. Nagarjuna himself is said to have been an early adherent to Mahakala, whose ideology is associated with annihilation, skepticism, and the Madhyamaka. There is a very detailed iconography left behind in the form of Tibetan Thangka paintings10. From these and other similar works we know that the physical form of Mahakala is a frightening Tryambaka11, and that he can be summoned to make householders go homeless. He usually holds a Vajra knife, which is the right shape and size to sever a cucumber from its stalk and chop it into little circular pieces. That is an example of Raudrantaka (sublimation of Rudra) practice. Mahakala is not a terrorist, and does not die from self-immolation, nor from other-imposed immolation. If someone scares you with Nature’s Truth, but also treats you with great Vatsalya, then you might have met an avatar of Mahakala. His consort Mahakali is supposed to send an avatar to remind us of the Dali Lama’s saying ”My religion is kindness”. Again, it is probably better to listen to the Zen Teachers who often say things like ”Never make Mahakala/Mahakali” or ”Don’t even think about Mara’s sons and daughters”. After so many years Buddhism went all the way from India, to Thailand, Vietnam and

7HA HA HA HA HA! Another great lecture: https://www.youtube.com/watch?v=SUXnycMe8oU. 8Here is the story: http://buddhism.lib.ntu.edu.tw/FULLTEXT/JR-BH/bh117515.htm. 9For example, disgust at the codependent problems of householding and homelessness. 10For a lot of examples, see: https://www.himalayanart.org/. 11Thanks again for this word, Ajeet Kaur, very sexy voice, please keep singing! 169 Fig. 5.2.: Blue, six-armed Mahakala. Unless you are a Tibetan expert, this is exactly what you should not want to make. See: https://en.wikipedia.org/wiki/Mahakala, appropriated from public domain. 170 Fig. 5.3.: Fascimile of Sengai’s famous painting (Fair Use from [2]). What does it mean?

Sri Lanka, to China, Korea, Japan, and then to America12—not only through the eﬀorts of Gary Snyder [32]. Avalokiteshvara became Guan-Yin, became Kwan-Um, became Kan-On, and in 1954 Kan-On became the star of Kenji Mizoguchi’s beautiful post-war elegy Sansho the Bailiﬀ. Later and farther in spacetime, after crossing the Great Oceans, the Boddhisatva of All Compassion had a meeting with Jesus and Muhammad in the middle of America. I learned some beginner practices at KU, refused to do the formal sitting, and most sadly was too late on arrival to ever meet the famous painter (now Kintsugi maker?) Stephen Addiss. Later in time, I happened to get a copy of his Zen Art Box [2] at the Dickson Street Bookstore, and therein found a nice reprinting of the old (Circa 1820) Sengai painting with a circle, a triangle, and a square. After studying curve geometries relative to Ramanujan’s Mystery, that painting struck me to the very core. Over time, its meaning has become something of a Koan. Why would Rinzai Zen Master Sengai be interested in such advanced

12Mystery Woman: ”Don’t forget Cambodia” . . . later on . . . ”And other little Asian countries, etc. etc.” 171 Fig. 5.4.: Initial study for ”Rei-kaku Ichi-kaku Ni-kaku” ”Mirror of Zen” ”Interbeing”, authors own work. Note: Rei∼ zero ∼ soul. Signed∼ with a bird seal.∼ Almost Sumi Ink and food coloring dye, ha∼ ha ha!∼

concepts? No one knows too sure, nor even what the Sumi ink painting should be called. While ” ” or ”The Universe” may have been okay answers a century ago, sorry to say, 4 now these answers taste stale and useless. So that is it, no more bad language. The dissertation is finally over. It might never be published as a ”Professional” work. Some of the bosses claimed they fired me, they at least tried to, and had some success limiting my access to resources. Others praised my writing. Still others compared it to mud or to cow dung. Forget it all! Now we can try to find somewhere to hide from the killer virus and argue about what actually happened, how, and why, etc. Here is a long list of plausible titles for Sengai’s painting: ”Do not make Ichi-kaku or Ni-kaku”, ”Rei-kaku Ken-sho En-so”, ”En-so En-so En-so”; ”Do ∼ ∼ ∼ ∼ not make Genus two”, ”Genus one okay”, ”Shapes and Curves okay”,”Okay to take Kala”, ”Okay to take Akasha”, ”Okay to take Akashakala”, ”Okay to make Buddhaverse”, ”Three

172 wheels with no spokes”, but ”Do not desire promotion”, ”Do not make Tryambaka”, ”Do not make Mahakala/Mahakali”! ”Do not make Atomic Bombs”! ”Do not make Sarin gas”! ”Do not make Machine guns”! Somedays I feel pathetic compared to the rich, house-holding Zen Masters. Then I remember that nothing even matters anymore. The Judge we need probably already died in a terrorist attack or during one of the many wars we fought. Best case scenario, maybe time expired naturally. Sengai Gibon (1750-1837) lived in roughly the same spacetime as Haiku poet and loving son Kobayashi Issa (1763-1828). The other title for Sengai’s famous painting, which I do like, was given by Issa. A translation is ”Even with insects / some can sing / some can’t” [25]. Vietnamese War survivor Thich Naht Hanh says that ”we are all drawing the circle together” [42], and I believe this pronouncement to be True, even at the insect level. Of course we are worried if bad human behaviors completely decimate the environment, that it will ultimately become impossible to paint even one more circle, but we have a few more years left to try and save the ecology. In the future, I hope to remember to treat other people with kindness, and to try and decrease Murder Karma. I hope you enjoyed the dissertation. Now you have it, or maybe if you work a little more you could. Please tell everyone you know—This is not a secret teaching!

Disclaimer. This chapter was written as a Post Mortem analysis, from the Author’s individual perspective, after a successful defense on Nov. 20, 2020. These insights do not necessarily convey the thoughts and opinions of the Dissertation Committee, nor of University of Arkansas administrators, nor of any political or religious organization.

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180 A. Supplemental Materials

Considerably many supplemental materials have been prepared and published for sake of reproducibility and transparency. The ﬁrst two places to check are Wolfram Demonstration Project [101] and Online Encyclopedia of Integer Sequences [89]:

• https://demonstrations.wolfram.com/author.html?author=Brad+Klee

• https://oeis.org/search?q=author:bradley+klee

A chapter-by-chapter breakdown of the most relevant resources now follows.

Chapter 1. Table 5 is proven in one notebook, which is available through github,

• https://github.com/bradklee/Dissertation/tree/master/Prelude/notebooks

The ﬁle ”TableCheck.nb” contains all three algorithms ExpToODE, HyperellipticToODE, and DihedralToODE, which appear (in one nomenclature or another) earlier in the following publications at Wolfram Demonstrations,

• Approximating Pi with Trigonometric-Polynomial Integrals

• A Few More Geometries after Ramanujan

To emphasize the practical side of diﬀerential equations during the COVID-19 outbreak of 2020, we also published two more Demonstrations,

• Summer Insect Pandemics in the United States

• Kermack-McKendrick SIR Model

Chapter 2. A few exact solutions in terms of elliptic functions have also been published at Wolfram Demonstrations,

• Edwards’s Solution of Pendulum Oscillation

• Discrete and Continuous Quartic Anharmonic Oscillation 181 • Weierstrass Solution of Cubic Anharmonic Oscillation

A general exploration of isoperiodicity is also available at Wolfram Demonstrations,

• Isoperiodic Potentials via Series Expansion

The quartic stratum featured in Section 5 was also published at Wolfram Demonstrations,

• D4 Symmetric Stratum of Quartic Plane Curves

Chapter 3. The most relevant references are already listed under the ”Chapter 1” heading above. Additionally, Wolfram Demonstrations hosts an early, hackerish attempt to understand Creative Telescoping (now deprecated), and a Lattice Walk notebook for the

four Ramanujan series, and another interesting application of an EasyCT-type algorithm,

• Deriving Hypergeometric Picard-Fuchs Equations

• Four Hypergeometric Lattice Walks

• Algebraic Family of Trefoil Curves

Chapter 4. Aside from various Demonstrations listed above, there is one more of interest and relevance,

• Semiclassical Quantization for Asymmetric Rigid Rotors

However, buyer beware, this notebook is not optimized per the text of Chapter 4. The basic idea of Chapter 4 was also given as a presentation at the International Symposium for Molecular Spectroscopy, and the slides are now available online,

• https://www.ideals.illinois.edu/handle/2142/104330

The annihilators for the Octahedral and Icosahedral geometries were originally published at OEIS,

• http://oeis.org/A318245 and http://oeis.org/A318495

Anything Else? The author has even more proof notebooks, which can be made available on request. Just send an email to [email protected]. 182 Bradley J. Klee

Contact Voice: (913)708-0783 Information E-mail: [email protected], [email protected]

Research Geometric physics, periods, creative telescoping, semi-classical quantum mechanics. Entomological Interests physics, insect population dynamics, talking insects. History of science and knowledge, West meets East, Buddhism, the Enlightenment debate. . .

Education University of Arkansas, Fayetteville, Arkansas USA Ph.D. in Mathematical Physics, November 20, December 2020 Dissertation: “An Update on the Computational Theory of Hamiltonian Period Functions” • Advisors: William G. Harter, Daniel Kenneﬁck, Salvador Barraza-Lopez, Edmund Harriss • University of Kansas, Lawrence, Kansas USA B.S. Physics, May 2010

Honors and University of Arkansas, Doctoral Academy Fellowship, 2013-2017 Awards University of Kansas, University Scholar, miscellaneous other scholarships, 2005-2010

Academic University of Arkansas, Fayetteville, Arkansas USA Experience Graduate Student Spring, 2013 - Winter, 2020 Dissertation research and writing, Ph.D. level coursework, and special projects

Laboratory Teaching Assistant Spring, 2013 - Spring, 2020 Experiment facilitation and supervision, project consultation, oﬃce hours, and grading

Papers in Klee, Bradley J. Prelude to a Well-Integrable Function Theory preparation Klee, Bradley J. An Alternative Theory of Simple Pendulum Libration

Klee, Bradley J. Geometric Interpretation of a few Elliptic Integrals

Klee, Bradley J. Developing the Vibrational-Rotational Analogy

Conference Klee, Bradley J. An Update on the Theory of Rotational Energy Surfaces, International Symposium Presentations on Molecular Spectroscopy, 2019, https://www.ideals.illinois.edu/handle/2142/104330

Supplemental Dissertation source, Github, https://github.com/bradklee/Dissertation Materials Mathematica computable documents, Wolfram Demonstrations Project, https://demonstrations.wolfram.com/author.html?author=Brad+Klee

Miscellaneous Integer Sequences, Online Encyclopedia of Integer Sequences, https://oeis.org/search?q=author:bradley+klee

Insect Observations, iNaturalist, https://www.inaturalist.org/observations?user id=bradklee