Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

TL;DR

This paper explores geometric methods to determine when dynamical systems can be reformulated as Hamiltonian systems using Poisson structures, addressing topological and symmetry considerations.

Contribution

It provides new conditions and methods for Hamiltonization of dynamical systems on manifolds, extending previous results and applying to systems with symmetries and group actions.

Findings

Hamiltonian formulation exists under certain topological conditions.

A variant of Hojman's construction applies to systems with invariant metrics.

Solutions are provided for systems with low-dimensional torus symmetries.

Abstract

Some positive answers to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures in a geometric framework. We address this problem on orientable manifolds and by using decomposable Poisson structures. In the first case, the existence of a Hamiltonian formulation is ensured under the vanishing of some topological obstructions, improving a result of Gao. In the second case, we apply a variant of the Hojman construction to solve the problem for vector fields admitting a transversally invariant metric and, in particular, for infinitesimal generators of proper actions. Finally, we also consider the hamiltonization problem for Lie group actions and give solutions in the particular case in which the acting Lie group is a low-dimensional torus.