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.