Dynamical Systems and Poisson Structures

TL;DR

This paper explores the Hamiltonian and Poisson structures of dynamical systems in various dimensions, providing algorithms for constructing these structures and classifying systems based on invariants, demonstrating their super-integrability.

Contribution

It introduces algorithms to find Poisson structures for dynamical systems in ${\mathbb R}^3$ and ${\mathbb R}^n$, and proves all such systems are bi-Hamiltonian or super-integrable.

Findings

All dynamical systems in ${\mathbb R}^3$ are bi-Hamiltonian.

All dynamical systems in ${\mathbb R}^n$ are $(n-1)$-Hamiltonian.

Autonomous systems in ${\mathbb R}^n$ are super-integrable.

Abstract

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.