TY - JOUR

T1 - Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

AU - Avendaño-Camacho, Misael

AU - García-Mendoza, Claudio César

AU - Ruíz-Pantaleón, José Crispín

AU - Velasco-Barreras, Eduardo

N1 - Publisher Copyright:
© 2022, Institute of Mathematics. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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 geomet-ric 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 sec-ond 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.

AB - 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 geomet-ric 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 sec-ond 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.

KW - Hamiltonian formulation

KW - Poisson manifold

KW - first integral

KW - symmetry

KW - trans-versally invariant metric

KW - unimodularity

UR - http://www.scopus.com/inward/record.url?scp=85131084897&partnerID=8YFLogxK

U2 - 10.3842/SIGMA.2022.038

DO - 10.3842/SIGMA.2022.038

M3 - Artículo

AN - SCOPUS:85131084897

SN - 1815-0659

VL - 18

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

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

M1 - 038

ER -