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

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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 initial 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. In addition, we include two other physical features and examine their effects. One is to consider the effects of inertia on the particle dynamics. In that case, oscillations in particle positions are observed which, under weak damping, still allow the particles to find their equilibrium states. The other is to allow the particles to move away from the surface into the interior of domain, which we analyze for circles and spheres. For the case of the circle, we find that as the particles become too crowded on the surface, it is energetically favorable for some of them to move into the interior where they arrange themselves in a lattice.
Ellipse and Ellipsoid