Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions - W. I. Skrypnik (Advances in Mathematical Physics)
The equation of motion in ℝd 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.