A variational approach to frozen planet orbits in helium

Abstract

We present variational characterizations of frozen planet orbits for the helium atom in the Lagrangian and the Hamiltonian picture. They are based on a nonlocal Levi-Civita regularization in Barutello, Ortega and Verzini (2021) with different time reparametrizations for the two electrons and lead to nonlocal functionals. Within this variational setup, we deform the helium problem to one where the two electrons interact only by their mean values and use this to deduce the existence of frozen planet orbits.