Homographic solutions of the curved 3-body problem

TL;DR

This paper investigates special solutions in the curved 3-body problem, proving the existence and classification of homographic orbits and analyzing the stability of hyperbolic Eulerian solutions.

Contribution

It establishes the existence and complete classification of Lagrangian and Eulerian homographic orbits in the curved 3-body problem with equal masses.

Findings

Existence of Lagrangian and Eulerian homographic orbits

Complete classification of these orbits for equal masses

Hyperbolic Eulerian relative equilibria are unstable

Abstract

In the 2-dimensional curved 3-body problem, we prove the existence of Lagrangian and Eulerian homographic orbits, and provide their complete classification in the case of equal masses. We also show that the only non-homothetic hyperbolic Eulerian solutions are the hyperbolic Eulerian relative equilibria, a result that proves their instability.