## Date of Graduation

12-2020

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Physics (PhD)

## Degree Level

Graduate

## Department

Physics

## Advisor/Mentor

William G. Harter

## Committee Member

Daniel J. Kennefick

## Second Committee Member

Edmund O. Harriss

## Third Committee Member

Salvador Barraza-Lopez

## Keywords

Algebraic Goemetry, Creative Telescoping, Hamiltonian Mechanics, Semiclassical Mechanics

## 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.

## Citation

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

#### Included in

Dynamical Systems Commons, Numerical Analysis and Scientific Computing Commons, Quantum Physics Commons, Theory and Algorithms Commons