Superintegrability and higher order constants for classical and quantum systems

TL;DR

This paper demonstrates that a broad class of classical and quantum systems with specific potentials are superintegrable, possessing higher order constants of motion, thus extending the understanding of integrability in these systems.

Contribution

It generalizes previous results by showing higher order constants of motion exist for a wider class of potentials in classical mechanics, supporting conjectures about quantum superintegrability.

Findings

Classical systems exhibit higher order constants of motion as polynomials in momenta.

Quantum systems in the studied class have higher order differential symmetry operators.

The results support the conjecture that such systems are superintegrable.

Abstract

We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classical mechanics, there are higher order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.