Equilibrium, regular polygons, and Coulomb-type dynamics in different dimensions. (English) Zbl 1478.78020

Summary: The equation of motion in \(\mathbb{R}^d\) 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.


78A35 Motion of charged particles
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35Q60 PDEs in connection with optics and electromagnetic theory
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