Two-body problem on a sphere. Reduction, stochasticity, periodic orbits

TL;DR

This paper investigates the dynamics of two particles on a sphere with Newtonian interactions, reducing the system to analyze periodic orbits, integrability, and stochastic behavior.

Contribution

It introduces a reduction method for the two-body problem on a sphere and identifies notable periodic orbits, advancing understanding of its complex dynamics.

Findings

Reduction to two degrees of freedom enables detailed analysis.

Identification of remarkable periodic orbits.

Discussion of integrability and stochasticity in the system.

Abstract

We consider the problem of two interacting particles on a sphere. The potential of the interaction depends on the distance between the particles. The case of Newtonian-type potentials is studied in most detail. We reduce this system to a system with two degrees of freedom and give a number of remarkable periodic orbits. We also discuss integrability and stochastization of the motion.