Cendra, Hernán and Holm, Darryl D. and Hoyle, Mark J. W. and Marsden, Jerrold E. (1998) The Maxwell–Vlasov equations in Euler–Poincaré form. Journal of Mathematical Physics, 39 (6). pp. 31383157. ISSN 00222488. http://resolver.caltech.edu/CaltechAUTHORS:CENjmp98

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Abstract
Low's wellknown action principle for the Maxwell–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's principle for the Eulerian description of Low's action principle then casts the Maxwell–Vlasov equations into Euler–Poincaré form for right invariant motion on the diffeomorphism group of positionvelocity phase space, [openface R]6. Legendre transforming the Eulerian form of Low's action principle produces the Hamiltonian formulation of these equations in the Eulerian description. Since it arises from Euler–Poincaré equations, this Hamiltonian formulation can be written in terms of a Poisson structure that contains the Lie–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–Vlasov Poisson structure is known, whose ingredients are the Lie–Poisson bracket on the dual of the Lie algebra of symplectomorphisms of phase space and the Born–Infeld brackets for the Maxwell field. We discuss the relationship between these two Hamiltonian formulations. We also discuss the general Kelvin–Noether theorem for Euler–Poincaré equations and its meaning in the plasma context.
Item Type:  Article 

Additional Information:  ©1998 American Institute of Physics. (Received 22 September 1997; accepted 21 January 1998) We would all like to extend our gratitude to T. Ratiu for his time and invaluable input. Some early (unpublished) work by J. Marsden, P. Morrison, and H. Gumral on aspects of the problem addressed in this paper helped us in the formulation given here and is gratefully acknowledged. In addition, M. Hoyle would like to thank Dion Burns, Ion Georgiou, Don Korycansky, Shinar Kouranbaeva, Myung Kim and David Schneider for their time and the discussions which helped him understand this material. Work by D. Holm was conducted under the auspices of the US Department of Energy, supported (in part) by funds provided by the University of California for the conduct of discretionary research by Los Alamos National Laboratory. J. Marsden was partially supported by US Department of Energy Contract No. DEFG0395ER25251. 
Subject Keywords:  BOLTZMANNVLASOV EQUATION; PLASMA; MIXTURES; MOTION; PHASE SPACE; TRANSFORMATIONS; LAGRANGIAN FUNCTION; EQUILIBRIUM PLASMA; STABILITY; HAMILTONIANS; LIE GROUPS; plasma transport processes; plasma instability; Vlasov equation; Lie algebras; transforms 
Record Number:  CaltechAUTHORS:CENjmp98 
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:CENjmp98 
Alternative URL:  http://dx.doi.org/10.1063/1.532244 
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ID Code:  1918 
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Deposited On:  23 Feb 2006 
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