Canonical Hamiltonian mechanics of Hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics

While a microscopic system is usually governed by canonical Hamiltonian mechanics, that of a macroscopic system is often noncanonical, reflecting a degenerate Poisson structure underlying the coarse-grained phase space. Probing into symplectic leaves (local structures in a foliated phase space), we may be able to elucidate the order of transition from micro to macro. The Lagrangian guides our analysis. We formulate canonized Hamiltonian systems of Hall magnetohydrodynamics (HMHD) which have a hierarchized set of canonical variables; the simplest system is the subclass in which the ion vorticity and magnetic field have integral surfaces. Renormalizing the singularity scaled by the reciprocal Hall parameter (as the ion vorticity surfaces and the magnetic surfaces are set to merge), we delineate the singular limit to ideal magnetohydrodynamics (MHD). The formulated canonical equations will be useful in the study of ordered structures and dynamics (with integrable vortex lines) in HMHD and their singular limit to MHD, such as magnetic confinement systems, shocks or vortical dynamics.