Equal masses Eulerian relative equilibria on a rotating meridian of S^2

Abstract

Relative equilibria on a rotating meridian on $\mathbb{S}^2$ in equal-mass three-body problem under the cotangent potential are determined. We show the existence of scalene and isosceles relative equilibria. Almost all isosceles triangles, including equilateral, can form a relative equilibrium, except for the two equal arc angles $θ= π/2$. For $θ\in (0,2π/3)\setminus \{π/2\}$, the mid mass must be on the rotation axis, in our case, at the north or south pole of $\mathbb{S}^2$. For $θ\in (2π/3,π)$, the mid mass must be on the equator. For $θ=2π/3$, we obtain the equilateral triangle, where the position of the masses is arbitrary. When the largest arc angle $a_\ell$ is in $a_\ell\in (π/2,a_c)$, with $a_c=1.8124...$, two scalene configurations exist for given $a_\ell$.