On interrelations between divergence-free and Hamiltonian dynamics

Abstract

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field X with a global cross-section there exist some 4-dimensional symplectic manifoldM ⊃ M and a smooth Hamilton function H :M → R such that for some c ∈ R one gets M = {H = c} and the Hamiltonian vector field X H restricted on this level coincides with X. For divergence free vector fields with singular points such the extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M . We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such the linearization is possible for tori with Diophantine rotation numbers.