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Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Hamiltonian formulation, first integral, unimodularity, transversally invariant metric, symmetry.
Mots-clés : Poisson manifold 
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     author = {Misael Avenda\~no-Camacho and Claudio C\'esar Garc{\'\i}a-Mendoza and Jos\'e Crisp{\'\i}n Ruiz-Pantale\'on and Eduardo Velasco-Barreras},
     title = {Geometrical {Aspects} of the {Hamiltonization} {Problem} of {Dynamical} {Systems}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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Misael Avendaño-Camacho; Claudio César García-Mendoza; José Crispín Ruiz-Pantaleón; Eduardo Velasco-Barreras. Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a37/

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