Eulerian relative equilibria of the curved 3-body problems in 𝐒²

Abstract

We consider the gravitational motion of n n point particles with masses m 1 m_1 , m 2 m_2 , … \ldots , m n > 0 m_n>0 on surfaces of constant Gaussian curvature. Based on the work of Diacu and his co-authors, we derive the law of universal gravitation in spaces of constant curvature. Using the results, we examine all possible 3 3 -body configurations that can generate geodesic relative equilibria. We prove the existence of all acute triangle Eulerian relative equilibria and get a necessary and sufficient condition for the existence of obtuse triangle Eulerian relative equilibria. We also show that any three positive masses can generate Eulerian relative equilibria.