On Hamiltonian and Action Principle Formulations of Plasma Dynamics

Abstract

Abstract.A general discussion of Hamiltonian and action principle formulations for fliud and plasma models is given. A procedure, based on Hamilton's principle of mechanics but adapted for continua, for the construction of action principles for fliud and kinetic models is given. The transformation from action principles in terms of the Lagrangian variable description to the Eulerian variable description in terms of noncanonical Poisson brackets is described. Two examples are developed: ideal MHD and Braginskii's fliud model with gyroviscosity.

Ill-Posedness of Free Boundary Problem of the Incompressible Ideal MHD

“…We transform the system (2.1) into Lagrangian variables (e.g. [18]). We denote η = η(t, a) = (η 1 , η 2 ) the position of a fluid element or parcel at time t with a = (a 1 , a 2 ) being the fluid element label, defined to be the position of the fluid element at the initial time, a = η(0, a), but this is not necessarily always the case.…”

Derivation of the Hall and extended magnetohydrodynamics brackets

“…For Extended MHD, an Eulerian action principle was proposed by [20] which was subsequently generalized to a Eulerian-Lagrangian action by [21]. For recent overviews of action principle formulations of plasma models, we refer the reader to [22][23][24]. The noncanonical Hamiltonian formulations for these models can be found in the works of [1,5,6,[25][26][27][28].…”

“…The B ± are advected by v ± , whose Lagrangian equivalents we will callq ± ≡q − α ±qd , where the minus sign comes because u andq d differ by a sign (see (23)). Thus we have these fully expanded expressions for B ± , in analogy with (11):…”

Inertial magnetohydrodynamics

“…Here, we construct an action by employing a method for building actions that was described in [34]. An important feature of this method is the Eulerian closure principle (ECP) that insures the resulting theory can be written in terms of an Eulerian set of variables.…”