Three-body relative equilibria on $S^2$ II: Extended Lagrangian configurations

TL;DR

This paper extends the study of three-body relative equilibria on the sphere by developing a new geometric method to analyze Lagrangian configurations, leading to the discovery of new solution families.

Contribution

It introduces a novel geometric technique to analyze extended Lagrangian relative equilibria on the sphere, simplifying the inertia tensor analysis and revealing new configurations.

Findings

New families of Lagrangian configurations identified

A simplified inertia tensor for analyzing equilibria

Extended the understanding of three-body dynamics on spheres

Abstract

This is a natural continuation of our first paper \cite{pre}, where we develop a new geometrical technique which allow us to study relative equilibria on the two sphere. We consider a system of three positive masses on $\mathbb{S}^2$ moving under the influence of an generic attractive potential which only depends on the mutual distances among the masses. We reduce the problem of finding extended Lagrangian relative equilibria to the analysis of the inertia tensor, then we obtain a more manageable equivalent inertia tensor which allow us to find new families of Lagrangian configurations.