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

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 ${\mathcal{B}}_{av}$ 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 $(z_1,z_2)\in {\mathcal{C}}_{{\mathcal{B}}_{av}}$ and find the 1-to-1 correspondence between them and positive rational numbers ${\mathbb{Q}}_{+}$. \rm