Dynamics and Equilibria of N Point Charges on a 2D Ellipse or a 3D Ellipsoid

Abstract

We consider the so-called Thomson problem which refers to finding the equilibrium distribution of a finite number of mutually repelling point charges on the surface of a sphere, but for the case where the sphere is replaced by a spheroid or ellipsoid. To get started, we first consider the problem in two dimensions, with point charges on circles (for which the equilibrium distribution is intuitively obvious) and ellipses. We then generalize the approach to the three-dimensional case of an ellipsoid. The method we use is to begin with a random distribution of charges on the surface and allow each point charge to move tangentially to the surface due to the sum of all Coulomb forces it feels from the other charges. Deriving the proper equations of motion requires using a projection operator to project the total force on each point charge onto the tangent plane of the surface. The position vectors then evolve and find their final equilibrium distribution naturally. For the case of ellipses and ellipsoids or spheroids, we find that multiple distinct equilibria are possible for certain numbers of charges, depending on the starting conditions. We characterize these based on their total potential energies. Some of the equilibria found turn out to represent local minima in the potential energy landscape, while others represent the global minimum. We devise a method based on comparing the moment-of-inertia tensors of the final configurations to distinguish them from one another.