Mountain pass frozen planet orbits in the helium atom model

Abstract

We seek frozen planet orbits for the helium atom through an application of the mountain pass lemma to the Lagrangian action functional. Our method applies to a wide class of gravitational-like interaction potentials, thus generalising the results in Cieliebak, Frauenfelder, and Volkov [Ann. Inst. H. Poincaré C Anal. Non Linéaire 40 (2023), 379–455]. We also let the charges of the two electrons tend to zero and perform an asymptotic analysis to prove convergence to a limit trajectory having a collision-reflection singularity between the electrons.