Motion on a given surface: potentials producing geodesic lines as trajectories

Abstract

In the light of inverse problem of dynamics, we consider the motion of a material point on an arbitrary two-dimensional surface, submersed in OE3. We study twodimensional potentials which produce a mono-parametric family of geodesic lines as trajectories. We establish a new, non-linear partial differential condition for the potential function V = V(u, v). With the aid of this condition, we examine if a given potential produces a family of geodesic lines on a certain surface or not. On the other hand, we can check if a given family of regular orbits is indeed a family of geodesic lines on a certain surface and then find the potential function V = V(u, v) which gives rise to this family of orbits. Special cases are also studied and pertinent examples are worked out.