A new method to study relative equilibria on $\mathbb{S}^2$

Abstract

We develop a new geometrical technique to study relative equilibria for a system of $n$--positive masses, moving on the two dimensional sphere $\mathbb{S}^2$, under the influence of a general potential which only depends on the mutual distances among the masses. The big difficulty to study relative equilibria on $\mathbb{S}^2$, that we call $RE$ by short, is the absence of the center of mass as a first integral. We show that the two vanishing components of the angular momentum, for motions on $\mathbb{S}^2$, play the same role as the center of mass for motions on the Euclidean plane. From here we obtain that the rotation axis of a $RE$ is one of the principal axes of the inertia tensor. Conditions for have $RE$ and relations between the shape (given by the arc angles $\sigma_{ij}$ among the masses) and the configuration (given by the polar angles $\theta_k$ and $\phi_i - \phi_j$ in spherical coordinates) are shown. For $n=3$, we show explicitly the conditions to have Euler and Lagrange $RE$ on $\mathbb{S}^2$. As an application of our method we study the the equal masses case for the positive curved three body problem where we show the existence of scalene and isosceles Euler $RE$ and isosceles Lagrange $RE$.