Abstract:
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.
References