Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2022
• Автори: Avendaño-Camacho, Misael, García-Mendoza, Claudio César, Ruíz-Pantaleón, José Crispín, Velasco-Barreras, Eduardo
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2022
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/211630
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems. Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón and Eduardo Velasco-Barreras. SIGMA 18 (2022), 038, 29 pages

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.