A variational approach to periodic orbits in the $e^{-}Z^{2+}e^{-}$ Helium
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911448172593152 |
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| author | Ye, Zixuan |
| author_facet | Ye, Zixuan |
| contents | 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 |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13655 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A variational approach to periodic orbits in the $e^{-}Z^{2+}e^{-}$ Helium Ye, Zixuan Dynamical Systems Mathematical Physics 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 |
| title | A variational approach to periodic orbits in the $e^{-}Z^{2+}e^{-}$ Helium |
| topic | Dynamical Systems Mathematical Physics |
| url | https://arxiv.org/abs/2602.13655 |