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.