Three body relative equilibria on $\mathbb{S}^2$

TL;DR

This paper investigates the conditions for relative equilibria in a three-body problem on the sphere $ ext{S}^2$, identifying specific configurations such as Euler and Lagrange types under a general potential depending on mutual angles.

Contribution

It provides explicit criteria for relative equilibria on $ ext{S}^2$ and demonstrates the existence of various symmetric and scalene configurations for equal masses with cotangent potential.

Findings

Existence of scalene and isosceles Euler relative equilibria.

Existence of isosceles and equilateral Lagrange relative equilibria.

Derived explicit conditions for relative equilibria on $ ext{S}^2$.

Abstract

We study relative equilibria ($RE$ in short) for three-body problem on $\mathbb{S}^2$, under the influence of a general potential which only depends on $\cos\sigma_{ij}$ where $\sigma_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cos\sigma_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene and isosceles Euler $RE$, and isosceles and equilateral Lagrange $RE$.