Superintegrable systems on sphere

TL;DR

This paper explores superintegrable generalizations of the Kepler problem on the three-dimensional sphere, including magnetic monopoles and complex fields, providing classification of motions and isomorphisms, with potential extensions to Lobachevsky space.

Contribution

It introduces new superintegrable systems on the sphere, including magnetic and generalized fields, and classifies their motions and symmetries.

Findings

Systems are integrable and superintegrable.

Existence of a vector integral analogous to the Laplace--Runge--Lenz vector.

Trajectory isomorphisms between planar and spatial motions.

Abstract

We consider various generalizations of the Kepler problem to three-dimensional sphere $S^3$, a compact space of constant curvature. These generalizations include, among other things, addition of a spherical analog of the magnetic monopole (the Poincar\'e--Appell system) and addition of a more complicated field, which itself is a generalization of the MICZ-system. The mentioned systems are integrable -- in fact, superintegrable. The latter is due to the vector integral, which is analogous to the Laplace--Runge--Lenz vector. We offer a classification of the motions and consider a trajectory isomorphism between planar and spatial motions. The presented results can be easily extended to Lobachevsky space $L^3$.