Spatial Relative Equilibria and Periodic Solutions of the Coulomb (n+1)-Body Problem

• Constantineau, Kevin ^[1] ; García-Azpeitia, Carlos ^[2] ; Lessard, Jean-Philippe ^[1]
  1. [1] McGill University

     McGill University

     Canadá

     
  2. [2] Universidad Nacional Autónoma de México

     Universidad Nacional Autónoma de México

     México

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 1, 2022
• Idioma: inglés
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• Resumen

  • 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 value μ=sk for k∈[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 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.