Spatial relative equilibria and periodic solutions of the Coulomb $(n+1)$-body problem

Abstract

We study a classical model for the atom that considers the movement of n charged particles of charge −1 (electrons) interacting with a fixed nucleus of charge µ > 0. We show that two global branches of spatial relative equilibria bifurcate from the n-polygonal relative equilibrium for each critical values µ = s k for k ∈ [2, ..., n/2]. In these solutions, the n charges form n/hgroups of regular h-polygons in space, where h is the greatest common divisor of k and n. Furthermore, each spatial relative equilibrium has a global branch of relative periodic solutions for each normal frequency satisfying some nonresonant condition. We obtain computer-assisted proofs of existence of several spatial relative equilibria on global branches away from the npolygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.