Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

TL;DR

This paper classifies all three-dimensional classical superintegrable systems in magnetic fields that are separable in Cartesian coordinates, focusing on quadratic integrals, and introduces methods for higher-order integrals.

Contribution

It provides a complete classification of quadratically superintegrable systems in magnetic fields separable in Cartesian coordinates, extending previous linear integral results.

Findings

Classified all minimally and maximally superintegrable systems with quadratic integrals.

Connected previous linear integral results to quadratic cases.

Presented methods for constructing systems with higher-order integrals.

Abstract

We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.