A variational approach to periodic orbits in the $e^{-}Z^{2+}e^{-}$ Helium

TL;DR

This paper employs variational methods and regularization techniques to analyze periodic orbits in a helium atom model, establishing a correspondence between these orbits and positive rational numbers.

Contribution

It introduces a variational framework for the helium atom with mean interaction and links periodic orbits to rational numbers, extending previous regularization approaches.

Findings

Counted the number of periodic orbits in the model.

Established a one-to-one correspondence between orbits and positive rationals.

Provided Lagrangian and Hamiltonian formulations for the solutions.

Abstract

In this article, we use variational approaches to describe generalized solutions $(q_1,q_2)$ and critical points $(z_1,z_2)$ of the action functional $\mathscr{B}_{av}$ for the Helium atom in the $e^{-}Z^{2+}e^{-}$ configuration with mean interaction, where $(q_1,q_2)$ and $(z_1,z_2)$ are related by a non-local Levi-Civita regularization introduced by Barutello, Ortega and Verzini. Additionally, we give the Lagrangian and the Hamiltonian formulations of the generalized solutions $(q_1,q_2)$ following the framework constructed by Cieliebak, Frauenfelder and Volkov. Finally, we count the number of periodic orbits $(z_1,z_2)\in \mathscr{C}_{\mathscr{B}_{av}}$ and find the 1-to-1 correspondence between them and positive rational numbers $\mathbb{Q}_{+}$. \rm