# Attracting and repelling 2-body problems on a family of surfaces of constant curvature

## TL;DR

This paper classifies rotational motions of two particles on a sphere with repelling interactions and investigates how the curvature of the surface influences the existence and stability of relative equilibria, including transitions through zero curvature.

## Contribution

It provides a geometric equivalence between repelling and attracting particles and analyzes the impact of curvature on two-body problem dynamics on surfaces of constant curvature.

## Key findings

- Classification of pure rotational motion on a sphere with repelling potential
- Analysis of relative equilibria as curvature varies, including stability behavior
- Clarification of curvature's role in existence and stability of equilibria

## Abstract

We first provide a classification of the pure rotational motion of 2 particles on a sphere interacting via a repelling potential. This is achieved by providing a simple geometric equivalence between repelling particles and attracting particles, and relying on previous work on the similar classification for attracting particles. The second theme of the paper is to study the 2-body problem on a surface of constant curvature treating the curvature as a parameter, and with particular interest in how families of relative equilibria and their stability behave as the curvature passes through zero and changes sign. We consider two cases: firstly one where the particles are always attracting throughout the family, and secondly where they are attracting for negative curvature and repelling for positive curvature, interpolated by no interaction when the curvature vanishes. Our analysis clarifies the role of curvature in the existence and stability of relative equilibria.