Frozen Planet Orbits for the n-Electron Atom

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Abstract
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.
Funding Information
  • Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni
  • Ministero dell'Università e della Ricerca (PRIN 2022FPZEES)
  • Ministero dell’Istruzione, dell’Università e della Ricerca (PRIN 20227HX33Z)

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