A constant of motion in 3D implies a local generalized Hamiltonian structure

TL;DR

This paper shows that any 3D nonautonomous ODE system with a constant of motion can be associated with a Poisson structure through a volume-preserving diffeomorphism, revealing a local generalized Hamiltonian framework.

Contribution

It establishes a method to construct a Poisson structure for 3D systems based solely on the existence of a constant of motion, expanding Hamiltonian theory applicability.

Findings

Poisson structure can be associated to 3D vector fields with a constant of motion.

Diffeomorphism preserves orientation and volume in phase space.

Provides a local generalized Hamiltonian structure for such systems.

Abstract

We demonstrate that a Poisson structure can always be associated to a general nonautonomous 3D vector field of ODEs by means of a diffeomorphism that preserves both the orientation and the volume of phase-space. The only prerequisite is the existence of one constant of motion.