Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

Chengle Peng; Xiudi Tang

Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

Symplectic Geometry 2025-11-07 v2

Authors: Chengle Peng , Xiudi Tang

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.

Keywords

spherical pendulum action-angle coordinates integrable system symmetric quadratic potential elliptic integrals monodromy

Cite

@article{arxiv.2509.04207,
  title  = {Action-angle coordinates of spherical pendulums with symmetric quadratic potentials},
  author = {Chengle Peng and Xiudi Tang},
  journal= {arXiv preprint arXiv:2509.04207},
  year   = {2025}
}

Comments

19 pages, 2 figures. Some calculation errors fixed. Closed form of action coordinates of symmetric quadartic and general cases added

Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

Abstract

Keywords

Cite

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