In collaboration with David Dritschel (Univ. of St-Andrews, Scotland), Rodrigo Schaefer (IM, UFRJ/UPC, Barcelone, Catalunya, Spain).
In the basic courses of mechanics a first approach to the central forces and, in particular, to the gravitational force, is made through Newton's laws and the expression of the Newtonian gravitational force
F=G m1 m2 / r^2.
It is well to remember that the 1 / r^2 dependence on the expression of force is a due contribution to Hooke's experiences in (see, among others, Arnold's book "Huygens and Barrow, Newton and Hooke"). In this approach the gravitational potential U(r) (F(x)=-\nabla_x U) is derived from the knowledge of the force.
How to find the expression of gravitational force when studying the mass dynamics in other geometries? For example 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:
We propose a formulation of the dynamics directly in the intrinsic geometry of the surface and that uses fundamental solutions of the equation of the gravitational field. We show how the equations of gravitational dynamics are closely linked to those of electric charges and to the dynamics of point vortices.
Furthermore, we shall show how known laws, such as Kepler's laws and some mechanics axioms (Newton's Laws), may depend on the geometry of the space, i.e. they are not universal properties.