Action-angle coordinates of spherical pendulums with symmetric quadratic potentials

2025/09/04 by Peng, Chengle, Xiudi Tang, Tang, Xiudi
Mathematics · Physics and Astronomy · #53D20 #70H06 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2509.04207

openalex publication_date 2025/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 = z², we represent its action-angle coordinates in terms of elliptic integrals, and calculate the monodromy.

Citations

• Symplectic geometry and spectral properties of classical and quantum coupled angular momenta
• Constructing integrable systems of semitoric type
• On global action‐angle coordinates

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