Dynamical systems and Poisson structures

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Guerses M., Guseinov G. S., ZHELTUKHIN K.

JOURNAL OF MATHEMATICAL PHYSICS, vol.50, no.11, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 11
  • Publication Date: 2009
  • Doi Number: 10.1063/1.3257919
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Middle East Technical University Affiliated: Yes


We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in R-3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in R-n are locally (n-1)-Hamiltonian. We give also an algorithm, similar to the case in R-3, to construct a rank two Poisson structure of dynamical systems in R-n. We give a classification of the dynamical systems with respect to the invariant functions of the vector field (X) over right arrow and show that all autonomous dynamical systems in R-n are super-integrable. (C) 2009 American Institute of Physics. [doi:10.1063/1.3257919]