Hamiltonian formulations for perturbed dissipationless plasma equations

Abstract

The Hamiltonian formulations for the perturbed Vlasov–Maxwell equations and the perturbed ideal magnetohydrodynamics (MHD) equations are expressed in terms of the perturbation derivative ∂F/∂ϵ≡[F,S] of an arbitrary functional F[ψ] of the Vlasov–Maxwell fields ψ=(f,E,B) or the ideal MHD fields ψ=(ρ,u,s,B), which are assumed to depend continuously on the (dimensionless) perturbation parameter ϵ. Here, [ , ] denotes the functional Poisson bracket for each set of plasma equations and the perturbation action functional S is said to generate dynamically accessible perturbations of the plasma fields. The new Hamiltonian perturbation formulation introduces a framework for functional perturbation methods in plasma physics and highlights the crucial roles played by polarization and magnetization in Vlasov–Maxwell and ideal MHD perturbation theories. One application considered in this paper is a formulation of plasma stability that guarantees dynamical accessibility and leads to a natural generalization to higher-order perturbation theory.