The Euler-Lagrange Cohomology and General Volume-Preserving Systems

TL;DR

This paper introduces Euler-Lagrange cohomology groups on symplectic manifolds and presents a cohomological framework for general volume-preserving systems, unifying various mechanics including Hamiltonian and Nambu systems.

Contribution

It develops a cohomological approach to volume-preserving systems, revealing a new 2-form analogous to the Hamiltonian and connecting different mechanics frameworks.

Findings

Established a cohomological formulation for volume-preserving equations.

Identified a 2-form playing a role similar to the Hamiltonian.

Linked the approach to Nambu mechanics and Feng-Shang systems.

Abstract

We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, \omega)$ and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian $H$ are included as a special case with the 2-form $\frac{1}{n-1} H \omega$. It is studied the other volume preserving systems on $({\cal M}^{2n}, \omega)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics.