We briefly introduce the conception on Euler-Lagrange cohomology groups on a
symplectic manifold (M2n,ω) 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 n−11​Hω. It is studied the other volume preserving
systems on (M2n,ω). It is also explored the relations between
our approach and Feng-Shang's volume-preserving systems as well as the Nambu
mechanics.Comment: Plain LaTeX, use packages amssymb and amscd, 15 pages, no figure