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 $\theta = \pi/2$. For $\theta\in (0,2\pi/3)\setminus \{\pi/2\}$, the mid mass must be on the rotation axis, in our case, at the north or south pole of $\mathbb{S}^2$. For $\theta\in (2\pi/3,\pi)$, the mid mass must be on the equator. For $\theta=2\pi/3$, we obtain the equilateral triangle, where the position of the masses is arbitrary. When the largest arc angle $a_\ell$ is in $a_\ell\in (\pi/2,a_c)$, with $a_c=1.8124...$, two scalene configurations exist for given $a_\ell$.