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Theses and Dissertations
12-2020
An Update on the Computational Theory of Hamiltonian Period Functions
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
by
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, J¨org 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)
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,