Applied and Computational Mathematics Seminar

Seminar Details

Speaker:
Stefanella Boatto
Affiliation:
Universidade Federal do Rio de Janeiro (Brazil) & INRIA-CetraleSupélec (France)
Title:
Vortex and mass dynamics over surfaces, Maxwell laws & the axioms of mechanics
Time:
3PM Wednesday, 11 October, 2017
Location:
SCN 1.25, O'Brien Centre for Science (North)

In collaboration with David Dritschel (Univ. of St-Andrews, Scotland), Rodrigo Schaefer (IM, UFRJ/UPC, Barcelone, Catalunya, Spain).

Abstract

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:

  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.

References

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