Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 038, 29 pages      arXiv:2103.00458      https://doi.org/10.3842/SIGMA.2022.038

Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón and Eduardo Velasco-Barreras
Departamento de Matemáticas, Universidad de Sonora, México

Received March 02, 2021, in final form May 10, 2022; Published online May 20, 2022

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.

Key words: Hamiltonian formulation; Poisson manifold; first integral; unimodularity; transversally invariant metric; symmetry.

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