A time-dependent energy-momentum method

TL;DR

This paper extends the energy-momentum method to non-autonomous Hamiltonian systems with symmetries, introducing a new stability analysis framework for relative equilibria in time-dependent settings.

Contribution

It generalizes the concept of relative equilibrium to non-autonomous systems and applies foliated Lie systems for stability analysis, broadening the scope of Hamiltonian stability methods.

Findings

Provides conditions for stability of relative equilibria in non-autonomous systems.

Reduces non-autonomous Hamiltonian systems using Marsden-Weinstein theorem.

Applies the framework to mechanical systems including rigid bodies.

Abstract

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of non-autonomous Hamiltonian systems are described via foliated Lie systems, which opens a new field of application of such differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden-Weinstein theorem and we provide conditions ensuring the stability of the projection of relative equilibrium points to the reduced space. As an application, we study the stability of relative equilibrium points for a class of mechanical systems, which covers rigid bodies as a particular instance.