Chinese Acad Sci, Inst High Energy Phys, Beijing 100039, Peoples R China
; Chinese Acad Sci, Inst Theoret Phys, Beijing 100080, Peoples R China
; Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
; Capital Normal Univ, Dept Math, Beijing 100037, Peoples R China
We briefly introduce the concept of Euler-Lagrange cohomology groups on a symplectic manifold (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 two-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 two-form 1/n-1 1 1 Homega. The other volume preserving systems on (M-2n ,omega) are studied. The relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics are also explored.