On interrelations between divergence-free and Hamiltonian dynamics

TL;DR

This paper explores the deep connections between divergence-free vector fields and Hamiltonian systems on 3D manifolds, establishing conditions for their equivalence, analyzing invariant tori, and discussing linearization possibilities under analyticity assumptions.

Contribution

It provides a rigorous framework linking divergence-free flows with Hamiltonian dynamics, including conditions for global and local reductions, and examines invariant tori and linearization in analytic cases.

Findings

Existence of a symplectic extension for divergence-free fields with a global cross-section.

Identification of invariant 2-tori with smooth invariant forms.

Linearization on invariant tori is possible in the analytic case with Diophantine rotation numbers.

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 manifold $\tilde{M}\supset M$ and a smooth Hamilton function $H: \tilde{M}\to \mathbb R$ such that for some $c\in \mathbb R$ 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.