Hamiltonian Systems with Symmetry, Coadjoint Orbits and Plasma Physics

The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the Lie-Poisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the Maxwell-Vlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the Poisson-Vlasov equations are shown to be in Hamiltonian form relative to the Lie-Poisson bracket on the dual of the (finite dimensional) Lie algebra of infinitesimal canonical transformations. Then we write Maxwell’s equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic fields, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and two-fluid electrodynamics, can be written in Hamiltonian form using similar group theoretic techniques. ∗Research partially supported by the Miller Institute and DOE Contract DE-ATO3-82ER12097, and the Miller Institute. †Research partially supported by NSF grant MCS-81-01642. ‡Lecture delivered by R. Schmid.