University of Arkansas, Fayetteville ScholarWorks@UARK Theses and Dissertations 12-2020 An Update on the Computational Theory of Hamiltonian Period Functions Bradley Joseph Klee University of Arkansas, Fayetteville Follow this and additional works at: https://scholarworks.uark.edu/etd Part of the Dynamical Systems Commons, Numerical Analysis and Scientific Computing Commons, Quantum Physics Commons, and the Theory and Algorithms Commons 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.