Periodic orbits for an infinite family of classical superintegrable systems

TL;DR

This paper proves that all bounded trajectories in a specific two-dimensional classical system with a particular potential are closed for all rational and integer values of a parameter, supporting the conjecture that its quantum counterpart is superintegrable.

Contribution

It demonstrates that the classical system with the given potential has closed bounded trajectories for all rational and integer k, confirming a conjecture about its superintegrability.

Findings

All bounded trajectories are closed for all integer and rational k.

The period of motion is constant, T=π/(2ω), independent of k.

Supports the conjecture that the quantum version is superintegrable.

Abstract

We show that all bounded trajectories in the two dimensional classical system with the potential $V(r,\phi)=\omega^2 r^2+ \frac{\al k^2}{r^2 \cos^2 {k \phi}}+ \frac{\beta k^2}{r^2 \sin^2 {k \phi}}$ are closed for all integer and rational values of $k$. The period is $T=\frac{\pi}{2\omega}$ and does not depend on $k$. This agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable.