Dynamical Systems and Poisson Structures

We first consider the Hamiltonian formulation of $n=3$ systems in general and show that all dynamical systems in ${\mathbb{R}}^3$ are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We find the Poisson structures of a dynamical system recently given by Bender et al. Secondly, we show that all dynamical systems in ${\mathbb{R}}^n$ are $(n-1)$-Hamiltonian. We give also an algorithm, similar to the case in ${\mathbb{R}}^3$, to construct a rank two Poisson structure of dynamical systems in ${\mathbb{R}}^n$. We give a classification of the dynamical systems with respect to the invariant functions of the vector field $\vec{X}$ and show that all autonomous dynamical systems in ${\mathbb{R}}^n$ are super-integrable.