Regularization of the Kepler problem on the Sphere

TL;DR

This paper explores multiple regularization techniques for the Kepler problem on the sphere, adapting classical methods and revealing their connections through transformations, thereby extending understanding of celestial mechanics in curved spaces.

Contribution

It introduces new regularization approaches for the Kepler problem on the sphere and relates them to Euclidean space methods via the gnomonic transformation.

Findings

Moser-type regularization adapted to spherical geometry

Ligon-Schaaf regularization applied to the spherical Kepler problem

Connections between spherical and Euclidean regularizations via transformations

Abstract

In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.