Three-Body Relative Equilibria on $\mathbb{S}^2$

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.