# Agenda

# Friday 2 February 2018

### Vortex and mass dynamics over surfaces: Maxwell laws, geometry and surface topology

#### Stefanella Boatto

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. It should be pointed out that, in the expression of the force, the 1/r^2 dependence was deduced from Hooke, based on many years of experiments (as a reference see, among others, Arnold’s book ”Huygens and Barrow, Newton and Hooke”). In such an approach the gravitational potential U (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 define a central force in an arbitrary geometry?

2) Given the distribution of matter on a given surface what is the fundamental equation for deducing the corresponding gravitational potential?

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. 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 dynamics.