On the generalized Hamiltonian structure of 3D dynamical systems

TL;DR

This paper explores the construction of Poisson structures in 3D dynamical systems with constants of motion, providing methods to derive these structures from solutions of linear PDEs and quadratures.

Contribution

It introduces a systematic approach to construct Poisson structures for 3D systems with one or two constants of motion, extending previous methods.

Findings

Poisson structures can be derived from linear PDE solutions when one constant of motion exists.

With two constants of motion, the problem reduces to quadrature, allowing more flexible structure functions.

The structure functions include arbitrary functions of the constants of motion, offering generalized solutions.

Abstract

The Poisson structures for 3D systems possessing one constant of motion can always be constructed from the solution of a linear PDE. When two constants of the motion are available the problem reduces to a quadrature and the structure functions include an arbitrary function of them.