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 . In this paper we study the extensions of the Euler and Lagrange relative equilibria ( for short) on the plane to the sphere. The on 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 and Euler . Another one is between the different types of the shapes of Lagrange . We prove that bifurcations between equilateral and isosceles Lagrange exist for the case of equal masses, and that bifurcations between isosceles and scalene Lagrange exist for the partial equal masses case.