Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Some geometric solutions to the problem of endowing a dynamical system with a Hamiltonian formulation are presented within the class of Poisson structures. 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, 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.