An n Center Problem as an Example of a Geodesic Billiard Map with Singularities: Poincare Maps and Dynamics

This paper deals with the motion of a particle of infinitesimal mass and moving under the gravitational attraction of n particles of equal mass and fixed in the vertices of a regular polygon. The complete reduction of this problem is done using its finite group of symmetries. We show that it is the lift of a flow which is topologically conjugate to a geodesic billiard problem with the Jacobi metric and also study numerically the corresponding billiard maps.