Superintegrable Lissajous systems on the sphere

TL;DR

This paper investigates superintegrable Lissajous-like systems on the sphere, demonstrating their symmetries, degeneracies, and constants of motion in both quantum and classical contexts, revealing their geometric and dynamical properties.

Contribution

It introduces a unified method to construct symmetries for both quantum and classical superintegrable Lissajous systems on the sphere.

Findings

Symmetries explain energy degeneracies in quantum systems

Constants of motion determine classical orbits and frequencies

The approach applies to both quantum and classical models

Abstract

A kind of systems on the sphere, whose trajectories are similar to the Lissajous curves, are studied by means of one example. The symmetries are constructed following a unified and straightforward procedure for both the quantum and the classical versions of the model. In the quantum case it is stressed how the symmetries give the degeneracy of each energy level. In the classical case it is shown how the constants of motion supply the orbits, the motion and the frequencies in a natural way.