Periodic orbits in time-dependent planar Stark-Zeeman systems
Urs Frauenfelder
TL;DR
The paper develops a variational framework for time-dependent Stark-Zeeman systems by applying the nonlocal loop-space blow-up to regularize periodic orbits and collisions. It derives a first-principles delay equation for the regularized dynamics and shows that critical points of the pulled-back action on the blown-up loop space correspond to collisional periodic solutions of the original ODE, via the complex squaring map. The results establish a precise correspondence that separates even and odd winding-number collisional solutions through untwisted and twisted loop spaces. This approach furnishes a robust tool for analyzing and continuing families of periodic orbits in time-dependent multi-body models, with potential to connect autonomous and non-autonomous regimes through homotopies.
Abstract
Time-dependent Stark-Zeeman systems describe the motion of an electron attracted by a proton subject to a magnetic and a time-dependent electric field. For instance the study of the dynamics of a gateway around the moon which is subject to the joint attraction of the moon, the earth and the sun leads to time-dependent Stark-Zeeman systems. In the time-dependent case there is no preserved energy. Therefore collisions cannot be regularized by blowing up the energy hypersurface. A new regularization technique of blowing up instead of the energy hypersurface the loop space was recently discovered by Barutello, Ortega, and Verzini. In this article we explain how this new regularization technique can be applied to the study of periodic orbits in time-dependent planar Stark-Zeeman systems. Since the regularization by blowing-up the loop space is nonlocal the regularized periodic orbits will not satisfy an ODE anymore but a delay equation.