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 $\mu >0$. We show that two global branches of spatial relative equilibria bifurcate from the n-polygonal relative equilibrium for each critical value $\mu =s_{k}$ for $k\in [2,\ldots ,n/2]$. In these solutions, the n charges form n/h-groups 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 n-polygonal relative equilibrium. Moreover, the nonresonant condition of the normal frequencies for some spatial relative equilibria is verified rigorously using computer-assisted proofs.