Three-body relative equilibria on $\mathbb{S}^2$ I: Euler configurations

TL;DR

This paper introduces a new geometric method to analyze three-body relative equilibria on a sphere, focusing on Euler configurations where bodies move along the same geodesic, enhancing understanding of such systems.

Contribution

It develops a novel geometric approach based on angular momentum properties to study relative equilibria of three bodies on a sphere, specifically analyzing Euler configurations.

Findings

Characterization of Euler configurations on $ ext{S}^2$

New geometric technique for relative equilibria analysis

Insights into three-body dynamics on curved surfaces

Abstract

Using the properties of the angular momentum, we develop a new geometrical technique to study relative equilibria for a system of $3$--bodies with positive masses, moving on the two sphere under the influence of an attractive potential depending only on the mutual distances among the bodies. With the above techniques we do an analysis of the relative equilibria for the case of three bodies when they are moving on the same geodesic (Euler configurations).