B. Khushalani

The problem of the stable configurations of N electrons on a sphere minimizing the potential energy of the system is related to the mathematical problem of the extremal configurations in the distance geometry and to the problem of the densest lattice packing of the congruent closed spheres. The arrangement of the points on a sphere in three-space leading to the equilibrium solutions has been of interest since 1904 when J. J. Thomson tried to obtain the stable equilibrium patterns of electrons moving on a sphere and subject to the electrostatic force inversely proportional to the square of the distance between them. Utilizing the theory of the point vortex motion on a sphere, Platonic polyhedral extremal configurations are obtained in this paper using numerical methods.

When inscribed inside a sphere of radius $R$, each of the five Platonic solids with a vortex of strength $\Gamma$ placed at each vertex gives rise to an equilibrium solution of the point vortex equations. In this paper, it will be shown how these equilibria can be used to generate families of periodic orbits on the sphere. These orbits are centered either around these equilibria or around more exotic equilibria, such as staggered ring configurations. Focussing on the cube as a generic case, four distinct families of periodic orbits made up of $24$ vortices (a $48$-dimensional system) are generated. These orbits bifurcate from the cube as each vertex is opened up with a splitting parameter $\theta$. The bifurcation from one orbit family to another is tracked by following the Floquet multipliers around the unit circle as the splitting parameter is varied.