Mass independent shapes for relative equilibria in the two dimensional positive curved three body problem

Toshiaki Fujiwara, Ernesto Perez-Chavela

Published 2023-11-05Version 1

In the planar three-body problem under Newtonian potential, it is well known that any masses, located at the vertices of an equilateral triangle generates a relative equilibrium, known as the Lagrange relative equilibrium. In fact, the equilateral triangle is the unique mass independent shape for a relative equilibrium in this problem. The two dimensional positive curved three-body problem, is a natural extension of the Newtonian three-body problem to the sphere $\mathbb{S}^2$, where the masses are moving under the influence of the cotangent potential. S.~Zhu showed that in this problem, equilateral triangle on a rotating meridian can form a relative equilibria for any masses. This was the first report of mass independent shape on $\mathbb{S}^2$ which can form a relative equilibrium. % In this paper, we show that, in addition to the equilateral triangle, three isosceles triangles on a rotating meridian, with $\theta_{ij}=\theta_{jk}=2^{-1}\arccos((\sqrt{2}-1)/2)$ for $(i,j,k)=(1,2,3)$, $(2,3,1)$, $(3,1,2)$ always form relative equilibria for any choice of the masses; where $\theta_{ij}$ are the angles between the two masses $i$ and $j$ seen from the centre of $\mathbb{S}^2$. Additionally we prove that, the equilateral and the above three isosceles relative equilibria are unique with this characteristic. We also prove that each relative equilibrium generated by a mass independent shape is not isolated from the other relative equilibria.