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

Zixuan Ye

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

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

Abstract

In this article, we use variational approaches to describe generalized solutions and critical points of the action functional for the Helium atom in the configuration with mean interaction, where and 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 following the framework constructed by Cieliebak, Frauenfelder and Volkov. Finally, we count the number of periodic orbits and find the 1-to-1 correspondence between them and positive rational numbers . \rm
Paper Structure (10 sections, 5 theorems, 98 equations, 2 figures)

This paper contains 10 sections, 5 theorems, 98 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Barutello-Ortega-Verzini regularization
  3. Variational approach to the $e^{-}Z^{2+}e^{-}$ Helium with mean interaction
  4. Variational characterization of generalized solutions
  5. The differential of $\mathscr{B}_{av}$
  6. Critical points of $\mathscr{B}_{av}$
  7. From critical points to generalized solutions
  8. From generalized solutions to critical points
  9. Hamiltonian Formulation
  10. Periodic orbits in mean $e^{-}Z^{2+}e^{-}$ Helium

Key Result

Theorem A

Under the Levi-Civita transformations $q_i(t)=z_i{(\tau_i(t))}^2(i=1,2)$ with time change $\tau_i(t)$ satisfying $\tau_i(0)=0$ and $\frac{dt}{q_i(t)}=\frac{d\tau_i(t)}{\parallel z_i\parallel ^2}$, critical points $(z_1,z_2)$ of the action functional $\mathscr{B}_{av}$ are in 4-to-1 correspondence wi .

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (14)

  • Theorem A
  • Theorem B
  • proof
  • Proposition 3.1
  • proof
  • proof
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 4 more