A variational approach to time-dependent planar two-center Stark-Zeeman systems

TL;DR

This paper develops a variational method using Birkhoff regularization to find periodic solutions, including collisions, in a complex time-dependent two-center Stark-Zeeman system relevant to celestial mechanics.

Contribution

It introduces a non-local regularized action functional for a time-dependent two-center system, enabling the study of collision solutions via variational methods.

Findings

Constructed a regularized action functional for the system.

Proved critical points correspond to periodic solutions including collisions.

Analyzed symmetry properties of the functional.

Abstract

We study periodic orbits in a time-dependent two-center Stark-Zeeman system, which models the motion of a charged particle attracted by two fixed Coulomb centers and subject to external magnetic and time-dependent electric fields. A motivating example is provided by the bicircular restricted four-body problem which investigates the motion of a massless particle, influenced by the Newtonian gravitational attraction of the earth, moon and periodically moving sun. Due to singularities at the Coulomb centers, standard local variational approaches fail. To overcome this, we employ the Birkhoff regularization map and construct a non-local regularized action functional on a blown-up loop space, following a recent method due to Barutello-Ortega-Vernizi \cite{BOV21}. We show that the critical points of this regularized action functional satisfy a certain second-order delay differential equation and correspond to periodic solutions of the original system, including collisions. Additionally, we examine the symmetric structure of the regularized functional.