  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.
