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 relativeequilibria ( \(RE\) 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 betweenLagrange \(RE\) and Euler \(RE\) . Another one is between the different types of the shapes of Lagrange \(RE\) . We prove thatbifurcations between equilateral and isosceles Lagrange \(RE\) existfor the case of equal masses, and that bifurcations between isosceles and scaleneLagrange \(RE\) exist for the partial equal masses case.
