New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory

Abstract

In this paper we describe a relatively simple and transparent method of obtaining collisionless drift-kinetic, gyrokinetic and more general theories, including local charge, energy and momentum conservation laws. An important feature of the new formalism is, contrary to present-day theories, the exact gauge invariance, thus avoiding certain inconsistencies. The present paper starts with the introduction and proof of the correctness of a Lagrangian for combined Maxwell-kinetic theories in general coordinates as concerns the particle motion. The kinetic part of it is formulated in Eulerian form by means of the equations of motion in the form of Hamilton–Jacobi's equation, used only as a tool, and Dirac's constraint theory. Charge and current densities automatically distinguish between ‘particle-like’ (guiding-centre), polarization and magnetization contributions. This formalism is applied to averaging coordinates derived by a method similar to Kruskal's. Certain properties of the averaging coordinates, according to the basic requirements imposed on them, can be used to obtain a gyroangle independent Lagrangian, from which one can obtain a Lagrangian for the combined Maxwell-kinetic theories in a reduced phase space that is applicable to situations in which one is not interested in the dependence on some kind of gyroangle describing the gyromotion, whose treatment can, however, easily be added. The basic perturbation theory, which aims at obtaining averaging phase-space coordinates, is done solely within the Kruskal formalism in which only the electric and magnetic fields appear, but not the corresponding potentials. This formalism provides, in particular, information about the allowed amplitudes of fluctuations. The results are later used to obtain approximate expressions for the Lagrange functions in the drift-kinetic and the gyrokinetic ordering. For the definition of certain approximations to the exact Lagrangian the central principle is the exact gauge invariance. The terms of the zeroth and first orders needed for such an approximation are given. For the drift-kinetic ordering Littlejohn's Lagrangian is readily rederived and hence the drift-kinetic theory as obtained and investigated by the present authors in some previous work. Conservation laws and their mathematical structures corresponding to the gyroangle-independent Lagrangian are obtained in a following paper. There, in particular, a detailed description of how to obtain variations of gyroangle averaged quantities in the reduced phase space is given, and it is explained how these variations are used in a modified form of Noether's theory which observes exact gauge invariance.