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<p>We investigate periodic trajectories in a classical system consisting of multiple mutually repelling electrons constrained to move along a half-line, with an attractive nucleus fixed at the origin. Adopting a variational framework, we seek critical points of the associated Lagrangian action functional using a modified Lusternik-Schnirelmann theory for manifolds with boundary. Furthermore, in the limit where the electron charges vanish, we demonstrate that frozen planet orbits converge to segments of a brake orbit in a Kepler-type problem, thereby drawing a strong analogy with Schubart orbits in the gravitational <i>N</i>-body problem.</p>

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Frozen Planet Orbits for the n-Electron Atom

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月30日

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  • Stefano Baranzini,
  • Gian Marco Canneori,
  • Susanna Terracini

摘要

We investigate periodic trajectories in a classical system consisting of multiple mutually repelling electrons constrained to move along a half-line, with an attractive nucleus fixed at the origin. Adopting a variational framework, we seek critical points of the associated Lagrangian action functional using a modified Lusternik-Schnirelmann theory for manifolds with boundary. Furthermore, in the limit where the electron charges vanish, we demonstrate that frozen planet orbits converge to segments of a brake orbit in a Kepler-type problem, thereby drawing a strong analogy with Schubart orbits in the gravitational N-body problem.