Revealing the phase space structure of Hamiltonian systems using the action

TL;DR

This paper demonstrates how the Maupertuis' action scalar field can effectively reveal the phase space structure of Hamiltonian systems, aiding in identifying key dynamical features like KAM islands and unstable orbits.

Contribution

It introduces a novel scalar field method based on the action to visualize and analyze phase space structures in Hamiltonian systems, validated through comparison with Poincaré maps.

Findings

The action scalar field reveals phase space objects such as unstable orbits and KAM islands.

Patterns in the scalar field correspond to different dynamical behaviors.

The method effectively complements traditional Poincaré map analysis.

Abstract

In this work, we analyse the properties of the Maupertuis' action as a tool to reveal the phase space structure for Hamiltonian systems. We construct a scalar field with the action's values along the trajectories in the phase space. The different behaviour of the trajectories around important phase space objects like unstable periodic orbits, their stable and unstable manifolds, and KAM islands generate characteristic patterns on the scalar field constructed with the action. Using these different patterns is possible to identify the skeleton of the phase space and understand the dynamics. Also, we present a simple argument based on the conservation of the energy and the behaviour of the trajectories to understand the values of their actions. In order to show how this tool reveals the phase space structures and its effectiveness, we compare the scalar field constructed with the actions with Poincare maps for the same set of initial conditions in the phase space of an open Hamiltonian system with 2 degrees of freedom.