Global realisation of magnetic fields as 1$\frac{1}{2}$D Hamiltonian systems

TL;DR

This paper explores how magnetic field line dynamics in certain 3D manifolds can be modeled as non-autonomous 1.5D Hamiltonian systems, providing conditions for such a representation and classifying the underlying geometric structures.

Contribution

It establishes conditions under which divergence-free vector fields on 3-manifolds can be identified as non-autonomous 1.5D Hamiltonian systems, linking topology, geometry, and Hamiltonian dynamics.

Findings

Field-line dynamics on certain 3-manifolds are equivalent to non-autonomous Hamiltonian systems.

All divergence-free fields transverse to a global Poincaré section are locally Hamiltonian.

Domains with disk or annulus Poincaré sections are diffeomorphic to solid or hollow tori, respectively.

Abstract

The paper reviews the notion of $n+\frac{1}{2}$D non-autonomous Hamiltonian systems, portraying their dynamics as the flow of the Reeb field related to a closed two-form of maximal rank on a cosymplectic manifold, and naturally decomposing into time-like and Hamiltonian components. The paper then investigates the conditions under which the field-line dynamics of a (tangential) divergence-free vector field on a connected compact three-manifold (possibly with boundary) diffeomorphic to a trivial fibre bundle over the circle can be conversely identified as a non-autonomous $1\frac{1}{2}$D Hamiltonian system. Under the assumption that the field is transverse to a global compact Poincar\'e section, an adaptation of Moser's trick shows that all such fields are locally-Hamiltonian. A full identification is established upon further assuming that the Poincar\'e sections are planar, which crucially implies (together with Dirichlet boundary conditions) that the cohomology class of the generating closed one-forms on each section is constant. By reviewing the classification of fibre bundles over the circle using the monodromy representation, it is remarked that as soon as the Poincar\'e section of an alleged field is diffeomorphic to a disk or an annulus, the domain is necessarily diffeomorphic to a solid or hollow torus, and thus its field-line dynamics can always be identified as a non-autonomous Hamiltonian system.