Generalized Thomson problem in arbitrary dimensions and non-euclidean geometries

• The confinement of equally charged particles in the Sd−1Sd−1-sphere is analyzed.
• The generalization of the so-called Thomson problem is performed.
• Compact structures appear in higher dimensions. d→∞d→∞ is addressed.
• New discrete systems are also studied: circumscribed and inscribed polygons.
• Non-Euclidean geometries are also considered.

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 SS and on the boundary of it ΩΩ (the Sd−1−Sd−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.