Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$

TL;DR

This paper explores the Kepler problem on curved spaces like the sphere and hyperbolic plane, deriving explicit solutions and analyzing conics in these geometries, unifying the Euclidean case as a special instance.

Contribution

It develops a unified framework for central potentials on spaces of constant curvature, providing explicit solutions and geometric insights for the Kepler problem in these settings.

Findings

Derived explicit orbit equations using two methods

Unified treatment of Kepler problem across different curvatures

Analyzed conic sections on curved spaces

Abstract

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature $\k$ as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for $\k>0$, or $\k<0$, respectively; in addition, the Euclidean case arises as the particular case $\k=0$. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods: first by direct integration and second by obtaining the $\k$-dependent version of the Binet's equation. The final part of the article, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.