Toshiaki Fujiwara

We study relative equilibria ($RE$) for the three-body problem on $\mathbb{S}^2$, under the influence of a general potential which only depends on $\cos\sigma_{ij}$ where $\sigma_{ij}$ are the mutual angles among the masses. Explicit conditions for masses $m_k$ and $\cos\sigma_{ij}$ to form relative equilibrium are shown. Using the above conditions, we study the equal masses case under the cotangent potential. We show the existence of scalene, isosceles, and equilateral Euler $RE$, and isosceles and equilateral Lagrange $RE$. We also show that the equilateral Euler $RE$ on a rotating meridian exists for general potential $\sum_{i\lt j} m_i m_j U(\cos \sigma_{ij})$ with any mass ratios.

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.