X iv : m at h - ph / 0 20 80 33 v 1 2 3 A ug 2 00 2 General Volume - Preserving Mechanical Systems

In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, ω). It is shown that every volume-preserving flow has some 2-forms acting the rôle of the Hamiltonian functions in the Hamiltonian mechanics and the ordinary Hamilton equations are included as a special case with a 2-form 1 n−1 H ω where H is the corresponding Hamiltonian. In classical mechanics, most of the systems can be described by the ordinary Hamilton equations q̇ i = ∂H ∂pi , ṗi = − ∂H ∂qi . (1) Email: [email protected] [email protected] [email protected]