Continuations and bifurcations of relative equilibria for the positive curved three body problem

TL;DR

This paper extends the study of the three body problem to spherical geometry, analyzing the continuation and bifurcations of relative equilibria, revealing new bifurcation phenomena between different configurations.

Contribution

It characterizes bifurcations of relative equilibria on the sphere, including between Euler and Lagrange types, and among different Lagrange configurations, for various mass distributions.

Findings

Bifurcations between Lagrange and Euler relative equilibria identified.

Existence of bifurcations between equilateral and isosceles Lagrange RE for equal masses.

Bifurcations between isosceles and scalene Lagrange RE for partial equal masses.

Abstract

The positive curved three body problem is a natural extension of the planar Newtonian three body problem to the sphere $\mathbb{S}^2$. In this paper we study the extensions of the Euler and Lagrange Relative equilibria ($RE$ in short) on the plane to the sphere. The $RE$ on $\mathbb{S}^2$ are not isolated in general. They usually have one-dimensional continuation in the three-dimensional shape space. We show that there are two types of bifurcations. One is the bifurcations between Lagrange $RE$ and Euler $RE$. Another one is between the different types of the shapes of Lagrange $RE$. We prove that bifurcations between equilateral and isosceles Lagrange $RE$ exist for equal masses case, and that bifurcations between isosceles and scalene Lagrange $RE$ exist for partial equal masses case.