All the Lagrangian relative equilibria of the curved 3-body problem have equal masses

TL;DR

This paper proves that in the curved 3-body problem on spaces of constant nonzero Gaussian curvature, all Lagrangian relative equilibria occur only when the three masses are equal, regardless of the class of equilibrium.

Contribution

It establishes that for all classes of Lagrangian relative equilibria in curved 3-space, the masses must be equal, extending classical results to curved geometries.

Findings

Lagrangian relative equilibria exist only for equal masses.

Three classes of equilibria are identified in curved spaces.

The result applies to both positive and negative curvature cases.

Abstract

We consider the 3-body problem in 3-dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.