Block regularization of the Kepler problem on surfaces of revolution with positive constant curvature

TL;DR

This paper studies the Kepler problem on curved surfaces of revolution with constant positive curvature, demonstrating superintegrability and identifying conditions for regularizable collision singularities, especially on spherical orbifolds.

Contribution

It establishes the superintegrability of the Kepler problem on such surfaces and characterizes when collision singularities are block-regularizable, extending classical results to curved geometries.

Findings

The system is maximally superintegrable with generalized Runge-Lentz vectors.

Collision singularities are regularizable on spherical orbifolds of revolution.

Conditions for block-regularizability depend on the surface's curvature and topology.

Abstract

We consider the Kepler problem on surfaces of revolution that are homeomorphic to $S^2$ and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces that lead to block-regularizable collision singularities. In particular we show that the singularities are always regularizable if the surfaces are spherical orbifolds of revolution with constant curvature.