A Surface of Section for Hydrogen in Crossed Electric and Magnetic Fields

Korana Burke,Kevin Mitchell

doi:10.3390/math6100185

Abstract

A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern the nonlinear dynamics of the system in question. In some cases, a global surface of section is relatively easily defined, but in other cases the definition is not trivial, and may not even exist. This is the case for the electron dynamics of a hydrogen atom in crossed electric and magnetic fields. In this paper, we demonstrate how one can define a surface of section and associated return map that may fail to be globally well defined, but for which the dynamics is well defined and continuous over a region that is sufficiently large to include the heteroclinic tangle and thus offers a sound geometric approach to studying the nonlinear dynamics.

Highlights

IntroductionGeometric structures lying within their phase spaces provide deep insights into their behaviors [1,2,3]
For many dynamical systems, geometric structures lying within their phase spaces provide deep insights into their behaviors [1,2,3]
This paper is organized as follows: In Section 2 we describe equations of motion for a hydrogenic electron in crossed fields; In Section 3 we present a prescription for finding an surface of section (SOS); Section 4 concludes by finding a periodic orbit and its corresponding tangle and turnstile, which are responsible for the ionization process

Summary

Introduction

Geometric structures lying within their phase spaces provide deep insights into their behaviors [1,2,3]. For two-degree-of-freedom Hamiltonian systems, a SOS is a two-dimensional surface in phase space and a heteroclinic/homoclinic tangle consists of one-dimensional stable and unstable manifolds within this surface In many cases it is challenging, or even impossible, to define a good SOS that captures all of the dynamics of the system in question. For the crossed fields case this simple construction fails to produce a good global SOS and, to the best of our knowledge, a good global surface of section does not appear in the literature This presents one of the major challenges to studying chaotic ionization in this case. This paper is organized as follows: In Section 2 we describe equations of motion for a hydrogenic electron in crossed fields; In Section 3 we present a prescription for finding an SOS; Section 4 concludes by finding a periodic orbit and its corresponding tangle and turnstile, which are responsible for the ionization process

Electron Equations of Motion

Construction of the Surface of Section

If the intersection is with one of the two left the SOS since

Visualizing a Periodic Orbit and Its Tangle

Conclusions