Integrable generalizations of oscillator and Coulomb systems via action-angle variables

TL;DR

This paper develops an action-angle variable framework for generalized oscillator and Coulomb systems on curved spaces, revealing their superintegrability and hidden constants of motion.

Contribution

It introduces a novel action-angle formulation for these systems on curved spaces, extending known models and demonstrating their superintegrability.

Findings

Constructed spherical and pseudospherical generalizations of known models

Proved superintegrability of the generalized systems

Identified hidden constants of motion

Abstract

Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable (N-1)-dimensional system. We present the action-angle formulation of such models in terms of theradial degree of freedom and the action-angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.