Geometrical Aspects of the Hamiltonization Problem of Dynamical Systems

Misael Avendaño-Camacho, Claudio César García-Mendoza, José Crispín Ruíz-Pantaleón, Eduardo Velasco-Barreras

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Abstract

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.

Bibliographical note

Funding Information:
We are very grateful to the anonymous referees for the observations and suggested improvements on various aspects of this work. This research was partially supported by the Mexican National Council of Science and Technology (CONACYT) under the grant CB2015 no. 258302 and the University of Sonora (UNISON) under the project no. USO315007338. J.C.R.P. thanks CONACyT for a postdoctoral fellowship held during the production of this work. E.V.B. was supported by FAPERJ grants E-26/202.411/2019 and E-26/202.412/2019.

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

Keywords

  • Hamiltonian formulation
  • Poisson manifold
  • first integral
  • symmetry
  • trans-versally invariant metric
  • unimodularity

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