Plane Pendulum and Beyond by Phase Space Geometry

TL;DR

This paper introduces a novel perturbation method for analyzing the dynamics of anharmonic oscillators, surpassing small angle approximations, with experimental validation and implications for higher-dimensional systems.

Contribution

It presents a new algorithm for deriving equations of motion and periods with arbitrary precision, and extends classical-quantum analogies for complex oscillatory systems.

Findings

Validated predictions with recursive data analysis

Derived equations of motion for anharmonic oscillators

Connected classical and quantum perturbation theories

Abstract

The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of a one-dimensional, parity-symmetric, anharmonic oscillator. A simple, novel algorithm produces the equations of motion and the period of oscillation to arbitrary precision. The Jacobian elliptic functions appear as a special case. Thrift experiment combined with recursive data analysis provides experimental verification of well-known predictions. Development of the quantum/classical analogy enables comparison of time-independent perturbation theories. Many of the useful notions herein generalize to integrable and non-integrable systems in higher dimensions.