Drift of slow variables in slow-fast Hamiltonian systems

TL;DR

This paper investigates how slow variables drift in slow-fast Hamiltonian systems, establishing conditions under which slow trajectories can shadow prescribed paths, thus revealing the influence of fast dynamics on slow variable evolution.

Contribution

It introduces a method to construct slow trajectories shadowing prescribed paths using action Hamiltonians derived from periodic orbits in the fast subsystem.

Findings

Existence of trajectories shadowing finite sequences of slow paths.

Conditions for hyperbolic periodic orbits and heteroclinic connections.

Framework for understanding slow variable drift in complex Hamiltonian systems.

Abstract

We study the drift of slow variables in a slow-fast Hamiltonian system with several fast and slow degrees of freedom. For any periodic trajectory of the fast subsystem with the frozen slow variables we define an action. For a family of periodic orbits, the action is a scalar function of the slow variables and can be considered as a Hamiltonian function which generates some slow dynamics. These dynamics depend on the family of periodic orbits. Assuming the fast system with the frozen slow variables has a pair of hyperbolic periodic orbits connected by two transversal heteroclinic trajectories, we prove that for any path composed of a finite sequence of slow trajectories generated by action Hamiltonians, there is a trajectory of the full system whose slow component shadows the path.