Expanding the simple pendulum's rotation solution in action-angle variables

TL;DR

This paper presents a novel method to explicitly expand the transformation to action-angle variables for the simple pendulum, simplifying the process by using Lie transforms instead of complex elliptic function expansions.

Contribution

It introduces a direct construction of the action-angle transformation that leverages Lie transforms, bypassing the need for intricate elliptic function expansions.

Findings

Explicit expansion of transformation achieved without elliptic functions

Lie transforms provide a natural framework for the expansion

Simplifies the analysis of the simple pendulum's Hamiltonian system

Abstract

Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions the transformation to action-angle variables may need to remain in implicit form. This is exactly the case of the simple pendulum, where in order to make explicit the transformation to action-angle variables one needs to resort to nontrivial expansions of special functions and series reversion. Alternatively, it is shown that the explicit expansion of the transformation to action-angle variables can be constructed directly, and that this direct construction leads naturally to the Lie transforms method, in this way avoiding the intricacies related to the traditional expansion of elliptic functions.