Dynamical manifestation of Hamiltonian monodromy

Abstract

PACS 45.50.-j -Dynamics and kinematics of a particle and a system of particles PACS 45.05.+x -General theory of classical mechanics of discrete systems

Dynamical manifestations of Hamiltonian monodromy

“…We show by computations that a passage around the monodromy circuit C carries particles to unexpected locations, so that the topology of the final loop in configuration space is different from that of the initial loop: the loop wraps around the forbidden region. In a previous short paper [1] we showed this behavior in a movie, and we urge the reader to read that reference and watch that movie before digging into the mathematical theory presented here. The purpose of this paper is to give a full analysis of this behavior.…”

“…Two ideas are central to this work [1]: (i) we define the perturbation implicitly by the resulting evolution of s ¼ ðm; EÞ and resulting perturbed trajectories and (ii) we study a family of trajectories, which we call 'particles', starting on a fundamental loopcð0Þ of a regular torus K ðm;EÞ .…”

“…The system we examine [1] is a classical particle of mass l moving in two dimensions q ¼ ðx; yÞ without friction inside a circular box of radius q max on a cylindrically symmetric potential-energy barrier VðqÞ ¼ À 1 2 aq 2 when q jqj 6 q max…”

“…Its main required property [1] is that it causes the particle to change its energy and angular momentum continuously in time, so that if we monitor EðtÞ and mðtÞ, we find that they go counterclockwise around a circuit C in the ðm; EÞ plane encircling the origin (0, 0) as shown in (2) is one of the simplest examples of systems with nontrivial monodromy [4]. In such a system, as will be explained in more detail later, if we relate continuously the coordinates on the neighboring regular tori with energy E and angular momentum m while ðm; EÞ follow a closed directed path C encircling (0, 0), and then compare final and initial coordinates after completing one tour and returning to the same torus, we will find that the final and initial coordinates do not match.…”

“…In this paper, we analyze the dynamical consequence of this phenomenon, which we called ''dynamical monodromy" [1]. In our case, the tori are connected dynamically by a (time-dependent) perturbation of the system.…”