Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions

Abstract

The equation of motion in ${ℝ}^d$ of $n$ generalized point charges interacting via the $s$ -dimensional Coulomb potential, which contains for $d=2$ a constant magnetic field, is considered. Planar exact solutions of the equation are found if either negative $n-1>2$ charges and their masses are equal or $n=3$ and the charges are different. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the infinite time for the considered equation without a magnetic field are also obtained. A condition permitting the existence of such solutions is proposed.