Three-body relative equilibria on $S^2$ II: Extended Lagrangian configurations

Abstract

This is a natural continuation of our first paper [6], where we develop a new geometrical technique which allow us to study relative equilibria on the two sphere. We consider a system of three positive masses on S 2 moving under the influence of an generic attractive potential which only depends on the mutual distances among the masses. We reduce the problem of finding extended Lagrangian relative equilibria to the analysis of the inertia tensor, then we obtain a more manageable equivalent inertia tensor which allow us to find new families of Lagrangian configurations.

Relative equilibria of mechanical systems with rotational symmetry

“…In section 4 we apply theorem 1 to obtain a complete and self-contained classification of the RE for the equal-mass spherical three-body problem. We would like to point out that this classification is not new, having recently been established in a series of preprints by Fujiwara and Pérez-Chavela [5][6][7][8][9] and the article [10]. In the final section we show how the energy-momentum method of [16] for assessing stability fits into our formalism.…”

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

“…In a recent work, we develop a systematic method to study relative equilibria on S 2 in the three-body problem [5,6]. This method is applicable to investigate the Euler configurations (relative equilibria where the three bodies are on a geodesic) and the extended Lagrange configurations (relative equilibria where the three bodies are not on a geodesic) with general masses.…”