Global realisation of magnetic fields as 1.5D Hamiltonian systems

This work is motivated by a common assumption in plasma physics that magnetic fields can be described by time-periodic non-autonomous Hamiltonian systems. We give a precise formulation of what it means for the dynamics of a vector field to be represented by a time-periodic non-autonomous Hamiltonian system and identify obstructions to the existence of such representations. Each obstruction is illustrated with examples. Our main result establishes necessary and sufficient conditions under which a vector field on a compact connected orientable three-manifold with boundary admits a Hamiltonian representation. In the physically relevant setting where the manifold is a solid or hollow torus embedded in Euclidean space, a corollary is that a magnetic field admits a Hamiltonian representation if and only if it possesses a global Poincaré section.