The Maxwell{endash}Vlasov equations in Euler{endash}Poincar{acute e} form
- Control and Dynamical Systems, California Institute of Technology 107-81, Pasadena, California91125 (United States)
- Los Alamos National Laboratory, Los Alamos, New Mexico87545 (United States)
- Department of Mathematics, University of California, Santa Cruz, California95064 (United States)
Low{close_quote}s well-known action principle for the Maxwell{endash}Vlasov equations of ideal plasma dynamics was originally expressed in terms of a mixture of Eulerian and Lagrangian variables. By imposing suitable constraints on the variations and analyzing invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we first transform this action principle into purely Eulerian variables. Hamilton{close_quote}s principle for the Eulerian description of Low{close_quote}s action principle then casts the Maxwell{endash}Vlasov equations into Euler{endash}Poincar{acute e} form for right invariant motion on the diffeomorphism group of position-velocity phase space, R{sup 6}. Legendre transforming the Eulerian form of Low{close_quote}s action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler{endash}Poincar{acute e} equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie{endash}Poisson bracket on the dual of a semidirect product Lie algebra. Because of degeneracies in the Lagrangian, the Legendre transform is dealt with using the Dirac theory of constraints. Another Maxwell{endash}Vlasov Poisson structure is known, whose ingredients are the Lie{endash}Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born{endash}Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin{endash}Noether theorem for Euler{endash}Poincar{acute e} equations and its meaning in the plasma context. {copyright} {ital 1998 American Institute of Physics.}
- OSTI ID:
- 624859
- Journal Information:
- Journal of Mathematical Physics, Vol. 39, Issue 6; Other Information: PBD: Jun 1998
- Country of Publication:
- United States
- Language:
- English
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