Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Symmetry Integrability and Geometry-Methods and Applications

article

Misael Avendaño-Camacho¹  Claudio César García-Mendoza¹  José Crispín Ruíz-PantaleónEduardo Velasco-Barreras¹ 

[1] Departamento de Matemáticas, Universidad de Sonora

关键词: Hamiltonian formulation;    Poisson manifold;    first integral;    unimodularity;    transversally invariant metric;    symmetry.;   

DOI  :  10.3842/SIGMA.2022.038

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown