Polynomial Poisson Algebras for Classical Superintegrable Systems with a Third Order Integral of Motion

TL;DR

This paper constructs polynomial Poisson algebras for 8 classical superintegrable systems with third order integrals of motion, analyzing their properties and classical trajectories.

Contribution

It introduces polynomial Poisson algebras for specific classical potentials with third order integrals, highlighting differences from quantum cases and trajectory periodicity.

Findings

All bounded trajectories are periodic.

Classical potentials sometimes differ from quantum counterparts as singular limits.

Polynomial Poisson algebras characterize these superintegrable systems.

Abstract

We present polynomial Poisson algebras for the 8 classical potentials in two-dimensional Euclidian space that separate in cartesian coordinates and allow a third order integral of motion. Some of the classical superintegrale potentials do not coincide with quantum ones, but are their singular limits. We show that all bounded trajectories in these potentials are periodic.