## 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.

### MSC:

 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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### References:

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