A variational approach to frozen planet orbits in helium

TL;DR

The paper develops a robust variational framework to study frozen planet orbits in the helium atom by regularizing billiard-like inner-electron collisions with nonlocal Levi-Civita-type transformations. It introduces two regularized functionals, $oxed{ ext{B}}_{av}$ and $oxed{ ext{B}}_{in}$, for mean and instantaneous electron interactions, and an interpolation $oxed{ ext{B}}_{r}$ linking them, establishing precise correspondences between critical points and generalized solutions. A key outcome is the existence of symmetric frozen planet orbits for any negative energy, proven via a mod 2 Euler characteristic in a Fredholm setting, with symmetry arguments extended through twisted- and symmetric-loop constructions. The work also provides a nonlocal Legendre transform to a Hamiltonian, yielding delay-type Hamilton equations that reproduce the same frozen planet orbits, thereby connecting Lagrangian and Hamiltonian nonlocal dynamics and inviting further study of indices and adiabatic-type limits in this context.

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 Levi-Civita regularization 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.

Theorems & Definitions (43)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Theorem 2.5: Barutello, Ortega and Verzini barutello-ortega-verzini
• Lemma 2.6
• Theorem 3.1: Generalized solutions with mean interaction
• Proposition 3.2
• Corollary 3.3
• Lemma 3.4