Noncanonical Hamiltonian field theory and reduced MHD

Marsden, J. E. and P. J. Morrison

Contemp. Math., 28, (1984), 133-150

Abstract:

Aspects of noncanonical Hamiltonian field theory are reviewed. Many systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular system of this type is considered, namely reduced magnetohydrodynamics (RMHD) which was derived for tokamak modelling. The notion of a Lie-Poisson bracket is reviewed; these are special Poisson brackets associated to lie groups. The RMHD equations are shown to be Hamiltonian for brackets closely related to the Poisson bracket of a semi-direct product group. The process by which this bracket may be derived from a canonical Lagrangian description by reduction is described.