Continuations and Bifurcations of Relative Equilibria for the Positively Curved Three-Body Problem

The positively 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 (\(RE\) for 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 the case of equal masses, and that bifurcations between isosceles and scalene Lagrange \(RE\) exist for the partial equal masses case.