Chaotic dynamics for the two-body problem on a sphere

We prove the existence of chaotic trajectories for the two body problem on a sphere. The trajectories we construct encounter near collisions and are similar to the second species solutions of Poincaré of the classical three body problem. The construction uses a general result on Lagrangian systems with Newtonian singularities of the potential, which is based on the method of an anti-integrable limit of Serge Aubry.