Gravitational and electrostatic potential fields and dynamics of non-spherical systems

Abstract

This thesis is devoted to several aspects of the n-body problem in the context of two models of interest: the gravitational n-body problem and the electrostatic n-body problem.

In the case of gravitational n-body problem, we study central configurations of three oblate bodies, the Hill approximation of the restricted four body problem with three oblate heavy bodies, and we find the equilibrium points of the Hill approximation and determine their linear stability. Also in the case of the gravitational n-body problem, we find equilibrium shapes of an irregular body, when the gravitational potential and the rotational potential balance each other. In particular, we find equilibrium dumbbell shapes.

In the context of the electrostatic n-body problem, we use variational methods to find approximate solutions of the Poisson-Boltzmann equation, representing the electrostatic potential produced by charged colloidal particles.

This research is motivated by applications to astrodynamics, dynamical astronomy and atomic force microscopy.