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

TL;DR

This paper classifies equal-mass three-body relative equilibria on a rotating meridian of the sphere, revealing conditions for scalene and isosceles configurations, including the existence of equilateral triangles and specific positional constraints.

Contribution

It provides a detailed analysis of relative equilibria on a sphere, identifying new configurations and the conditions under which they occur, especially for scalene and isosceles triangles.

Findings

Almost all isosceles triangles form relative equilibria, except for two equal arc angles.

The mid mass position depends on the arc angle, either on the pole or the equator.

Two scalene configurations exist for certain largest arc angles.

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 $a_\ell$.