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

TL;DR

This paper investigates spatial relative equilibria and periodic solutions in a Coulomb n-body model, revealing bifurcations from polygonal configurations and establishing existence through computer-assisted proofs.

Contribution

It introduces new bifurcation results for spatial relative equilibria in Coulomb systems and rigorously verifies conditions for periodic solutions using computational methods.

Findings

Bifurcation of spatial relative equilibria from n-polygonal configurations at specific charge values.

Existence of continuous branches of relative periodic solutions around these equilibria.

Rigorous computer-assisted verification of nonresonant frequency conditions.

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 values $\mu =s_{k}$ for $k\in \lbrack 2,...,n/2]$. 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 the 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.