Gluing Torus Families across a Singularity: The Lens Space for the Hydrogen Atom in Crossed Fields

Abstract

We demonstrate that topological information that can be extracted from periodic orbits in a near-integrable system can lead to a complete topological characterization of families of invariant 2-tori in terms of lens spaces. This approach ties in with the techniques we developed for classifying the tori in systems with more than two degrees of freedom. It therefore offers a general way to investigate families of invariant 2-tori in higher-dimensional Hamiltonian systems. §1. IntroductionPeriodic orbits are the simplest invariant objects in the phase space of a dynamical system. 1) Because they form the fundamental building blocks of the dynamics and because they are much simpler to calculate numerically than any more complicated structures, they are widely used as a tool to study the geometrical and dynamical structure of a multidimensional phase space. 2), 3) In a near-integrable system, the phase-space structure is dominated by a hierarchy of invariant tori of various dimensions. As we showed in a recent series of publications, 4) the existence of such a hierarchy, about which no detailed knowledge is required, can be used to impose an ordering principle upon the periodic orbits, and conversely, this ordering of periodic orbits allows one to obtain information about the higher-dimensional invariant tori that are not easily accessible through direct numerical methods.In the present paper we will demonstrate how the results of our earlier work can be extended into a complete topological characterization of families of invariant tori. Since we found the familiar Poincaré surface of section plots a somewhat unreliable tool to this end, we will analyze the singularities of Poincaré maps that lead to this failure, and we will show how the nature of the singularities is related to the topology of the family of tori under consideration.Because the present paper is focused on the topology of families of 2-tori, which exist in systems with two or more degrees of freedom, and on their representation in Poincaré plots, which are available only in two degrees of freedom, we will largely restrict our discussion to this case. Nevertheless, we will also show how these results can be generalized to higher-dimensional systems.

Ergodic edge modes in the 4D quantum Hall effect

“…1). 3 More generally, any rational ratio (7) leads to periodic trajectories in a (squashed) S 3 , generalizing the isotropic case so that each equipotential is a (covering of a) lens space [49], with distinct fibers again labelled smoothly by points in S 2 . By contrast, when ∆ is irrational, the motion of x(t) is ergodic in the sense that it fills its torus densely, regardless of the initial condition x(0).…”

Complete classification of qualitatively different perturbations of the hydrogen atom in weak near-orthogonal electric and magnetic fields

“…Here we mention only the most directly related work. A few more distant references are [11][12][13][14][15][16][17]. A thorough review of all works on other aspects of this problem (ionization, chaotic regime, purely quantum computations of low-lying states, etc) is beyond the scope of our work.…”