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 $a_c=\mathrm{1.8124...}$, two scalene configurations exist for given $a_{\ell }$.

20 pages, 5 figures