The restricted three body problem on surfaces of constant curvature

We consider a symmetric restricted three-body problem on surfaces Mκ2 of constant Gaussian curvature κ≠0, which can be reduced to the cases κ=±1. This problem consists in the analysis of the dynamics of an infinitesimal mass particle attracted by two primaries of identical masses describing elliptic relative equilibria of the two body problem on Mκ2, i.e., the primaries move on opposite sides of the same parallel of radius a. The Hamiltonian formulation of this problem is pointed out in intrinsic coordinates. The goal of this paper is to describe analytically, important aspects of the global dynamics in both cases κ=±1 and determine the main differences with the classical Newtonian circular restricted three-body problem. In this sense, we describe the number of equilibria and its linear stability depending on its bifurcation parameter corresponding to the radial parameter a. After that, we prove the existence of families of periodic orbits and KAM 2-tori related to these orbits.