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 \(\sum_{i<j}m_{i}m_{j}U(\cos\sigma_{ij})\) with any mass ratios.