Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

TL;DR

This work analyzes the spherical pendulum with a potential $V(z)$ as a Liouville-integrable system with a periodic Hamiltonian flow, deriving explicit solutions and constructing action-angle coordinates. It specializes to the symmetric quadratic case $V(z)=z^2$, expressing the joint flow and the action-angle coordinates in terms of elliptic integrals, and computing the monodromy associated with the focus-focus singularity. The results yield explicit period lattices and continuous action variables, highlighting the geometric structure (including a doubly pinched torus) and the nontrivial monodromy that arises in this natural mechanical system. The findings contribute to explicit symplectic invariants for a class of semitoric-like systems and provide a concrete example where action-angle coordinates can be written in closed form using classical elliptic functions.

Abstract

We study the spherical pendulum system with an arbitrary potential function $V = V (z)$, which is an integrable system with a first integral whose Hamiltonian flow is periodic. We give an explicit solution to this integrable system and then we compute its action-angle coordinates. In the special case where the potential function is symmetric quadratic like $V = z^2$, we represent its action-angle coordinates in terms of elliptic integrals, and calculate the monodromy.

Figures (1)

• Figure 1: The momentum map image and fibers.