Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

TL;DR

This paper establishes necessary conditions for the maximal super-integrability of specific classical potentials in constant curvature spaces, providing explicit examples and forms of additional polynomial first integrals of arbitrary high degree.

Contribution

It introduces necessary conditions for super-integrability in constant curvature spaces and demonstrates their sufficiency for certain homogeneous potentials, including explicit polynomial first integrals.

Findings

Necessary conditions for super-integrability are formulated.

Examples of potentials satisfying these conditions are provided.

Explicit polynomial first integrals of arbitrary degree are constructed.

Abstract

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$ as well as their equivalents on $S^2$ and $H^2$ for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which always can be chosen polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta.