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.

💡 Research Summary

The paper addresses a fundamental question in the geometric formulation of dynamical systems: how universally can a given system be expressed in Hamiltonian form, and how many distinct Hamiltonian structures can coexist. In the first part the authors focus on three‑dimensional autonomous systems. They prove that any vector field (X) on (\mathbb{R}^3) admits two independent Poisson tensors (J^{(1)}) and (J^{(2)}) of rank two, such that
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