Nqticanorucal Hamil To~ian Fl Eld Theory and Reduced ~lho

Aspects of noncanonical Hamiltonian field theory are reviewed. '·1any systems are Hamiltonian in the sense of possessing Poisson bracket structures, yet the equations are not in canonical form. A particular sys tem of thi s type is cons idered, namely reduced magnetohydrodynamics (RllHD) which was derived for tokamak modelling. The notion of a liePoisson bracket is reviewed; these are special Poisson brackets associated to Lie groups. The RI4iD equations are shown to be Hamiltonian for brackets closely related to the Poisson bracket of a semi-di rect product group. The process by which this bracket may be derived from a canonical Lagrangian description by reduction is described.