Abstract
We study the motion of the negative curved symmetric two and three center problem on the Poincare upper semi plane model for a surface of constant negative curvature kappa, which without loss of generality we assume kappa = -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 nonexistence 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.