In basic courses of mechanics a first approach to central forces and, in particular, to the gravitational force, is made through Newton’s laws and the expression of the Newtonian gravitational force. In such an approach the gravitational potential U (r)=k/r (F (x) = −grad(U) ) is derived from the knowledge of the force. How to find the expression of the gravitational force when studying the mass dynamics in other geometries? For examples on surfaces? We have the problem of not being able to perform two-dimensional experiments to measure the force between two bodies and therefore we must find the answer to the following : 1) How to generalize the notion of gravitational force to an arbitrary geometry? Various possible generalizations : the Hodge decomposition of vector fields and normal forms provides us with a possible generalisation 2) Given the distribution of matter on a given surface what is the fundamental equation for deducing the corresponding gravitational potential?
We propose a Maxwell-like formulation of the dynamics related to the definition of a central force and directly formulated in the intrinsic geometry of the surface. We show how the corresponding equations of gravitational dynamics are closely linked to those of electric charges and to the dynamics of point vortices. This allows us to take advantage in part of what we already being done for vortex dynamics and relative equilibria of vortices on surface of constante Gaussian curvature. Furthermore, we shall show how known laws may depend on the geometry of the space, i.e. they are not universal properties. Among other things, we show that in the plane the 2-body problem does not obeys to the known Kepler laws. For masses on an infinite cylinder we are able to observe topological effects in the mass dynamics.