Frozen Planet Orbits for the n-Electron Atom

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.