Hüseyin, Hüseyin ŞirinGuerses, MetinGuseinov, Gusein ShZheltukhin, KostyantynMathematics2024-07-052024-07-052009160022-24881089-765810.1063/1.32579192-s2.0-72249092277https://doi.org/10.1063/1.3257919https://hdl.handle.net/20.500.14411/1530Zheltukhin, Kostyantyn/0000-0002-1098-7369; Gurses, Metin/0000-0002-3439-3952We 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]eninfo:eu-repo/semantics/openAccess[No Keyword Available]ArticleQ35011WOS:000272755100015