Action-angle coordinates for the pendulum problem

TL;DR

This paper derives the action-angle coordinates for the planar pendulum using Jacobi elliptic functions and integrals, providing a canonical transformation applicable to both librating and rotating motions.

Contribution

It introduces a method to express the pendulum's action-angle coordinates explicitly via Jacobi elliptic functions and the Jacobi zeta function, enhancing analytical understanding.

Findings

Explicit formulas for action-angle coordinates using elliptic functions

Canonical transformation between physical and action-angle variables

Applicable to both librating and rotating pendulum motions

Abstract

The action-angle coordinates for the planar pendulum problem are expressed in terms of the Jacobi elliptic functions and integrals. In particular, we show that the Jacobi zeta function generates the canonical transformation from the pendulum coordinates $\vartheta$ and $p \equiv \partial\vartheta/\partial t$ to the action-angle coordinates $(J,\zeta)$ for both the librating pendulum and the rotating pendulum.