General Volume-Preserving Mechanical Systems

TL;DR

This paper derives the general equations for volume-preserving flows on symplectic manifolds, extending Hamiltonian mechanics by introducing 2-forms that generalize Hamiltonian functions, thus broadening the scope of volume-preserving dynamical systems.

Contribution

It introduces a unified framework for volume-preserving flows using 2-forms, generalizing Hamiltonian mechanics on symplectic manifolds.

Findings

Every volume-preserving flow can be generated by specific 2-forms.

Hamiltonian equations are a special case within this framework.

Provides a new perspective on volume-preserving dynamical systems.

Abstract

In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, \omega). It is shown that every volume-preserving flow has some 2-forms acting the role of the Hamiltonian functions in the Hamiltonian mechanics and the ordinary Hamilton equations are included as a special case with a 2-form \frac{1}{n-1} H \omega where H is the corresponding Hamiltonian.