The curved symmetric 2– and 3–center problem on constant negative surfaces.

We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature κ, which without loss of generality we assume κ = −1. Using this model, we first derive the equations of motion for the 2-and 3-center problems. We prove that for 2–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant y–axis, we prove that it is a center, but for the general two center problem it is unstable. For the 3–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the y–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation. [ABSTRACT FROM AUTHOR]