Hamiltonian structure of the Vlasov-Maxwell system in a curved background spacetime

Details Hamiltonian structure of the Vlasov-Maxwell system in a curved background spacetime Hamiltonian structure of the Vlasov-Maxwell system in a curved background spacetime

Nov 1993

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1993phrvd..48.4534k&link_type=abstract

Physical Review D (Particles, Fields, Gravitation, and Cosmology), Volume 48, Issue 10, 15 November 1993, pp.4534-4544

Physics

4

Relativity And Gravitation, Classical Electromagnetism, Maxwell Equations, Kinetic Theory

Scientific paper

Working in the context of an ADM splitting into space plus time, this paper identifies a Hamiltonian formulation, i.e., cosymplectic structure, for the Vlasov-Maxwell system, as formulated in a fixed, albeit curved, background spacetime. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial vector potential Ai, and the conjugate momentum Πi. This Hamiltonian formulation entails the identification of (i) Lie brackets , defined for functionals F[Ai,Πi,f] and G[Ai,Πi,f], and (ii) a Hamiltonian function H[Ai,Πi,f], so chosen that the equations of motion take the form ∂tF=, with ∂t a coordinate time derivative. This formulation is used to address the problem of stability of equilibria corresponding to time-independent electromagnetic fields in a spacetime admitting a timelike Killing field. An explicit expression is derived for the most general dynamically accessible perturbation δX≡(δAi,δΠi,δf), and it is shown that all equilibria are energy extremals with respect to such δX, i.e., δ(1)H[δX]≡0. The sign of the second variation δ(2)H, also computed, is thus directly related to the problem of linear stability: If δ(2)H[δX]>0 for all dynamically accessible perturbations, the equilibrium is guaranteed to be linearly stable. The existence of some perturbation δX for which δ(2)H[δX]<0 does not guarantee a linear instability. However, one does at least anticipate that equilibria admitting such perturbations will be nonlinearly unstable and/or unstable towards the effects of dissipation.

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