Generalized Thomson problem in arbitrary dimensions and non-euclidean geometries

TL;DR

This paper generalizes the Thomson problem to multiple dimensions and non-Euclidean geometries, analyzing how identical charged particles arrange themselves under Coulomb repulsion in various curved spaces.

Contribution

It introduces a generalized framework for the Thomson problem in arbitrary dimensions and geometries, revealing symmetry properties of optimal configurations.

Findings

Optimal configurations exhibit high symmetry across all cases.

Configurations depend on the dimension and geometry of the space.

The study extends classical Coulomb systems to non-Euclidean contexts.

Abstract

Systems of identical particles with equal charge are studied under a special type of confinement. These classical particles are free to move inside some convex region S and on the boundary of it $\Omega$ (the $S^{d-1}-$ sphere, in our case). We shall show how particles arrange themselves under the sole action of the Coulomb repulsion in many dimensions in the usual Euclidean space, therefore generalizing the so called Thomson problem to many dimensions. Also, we explore how the problem varies when non-Euclidean geometries are considered. We shall see that optimal configurations in all cases possess a high degree of symmetry, regardless of the concomitant dimension or geometry.