The Euler-Lagrange Cohomology and General Volume-Preserving Systems

Details The Euler-Lagrange Cohomology and General Volume-Preserving Systems The Euler-Lagrange Cohomology and General Volume-Preserving Systems

2003-04-08

arxiv.org/abs/hep-th/0304075v2

Mod.Phys.Lett. A18 (2003) 1911-1924

Physics

High Energy Physics

High Energy Physics - Theory

Plain LaTeX, use packages amssymb and amscd, 15 pages, no figures

Scientific paper

10.1142/S0217732303011708

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.

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