Covariant Lagrangian Methods of Relativistic Plasma Theory

Abstract

We obtain a covariant decomposition of the motion of a relativistic charged particle into parallel motion and perpendicular gyration, and transform to guiding-center coordinates using Lie transforms. The natural guiding-center Poisson bracket structure and Hamiltonian are derived. The guiding-center equations of motion are presented to one order higher than the usual drifts, and the correction to the gyromomentum is given. We then allow for eikonal wave perturbations to the Lagrangian action. We develop a manifestly gauge-invariant and covariant oscillation-center theory to arbitrarily high order, and thereby derive the relativistic ponderomotive Hamiltonian. We sum the guiding-center action over a distribution and add the Maxwell action to obtain the total action of a guiding-center plasma. Upon variation, this yields self-consistent covariant relativistic kinetic and field equations; from these we identify the guiding-center current density and the guiding-center magnetization. Noether's theorem then yields covariant conservation laws for the momentum-energy and the angular momentum of a relativistic guiding-center plasma; from these we identify the guiding-center stress-energy tensor and the guiding-center spin angular momentum tensor. Repeating this calculation for guiding/oscillation centers, we find that variation yields self-consistent relativistic kinetic and field equations for the plasma in the wave field, including the wave dispersion relation; from these we identify the wave magnetization and susceptibility. Noether's theorem then yields conservation laws for the guiding-center plasma in the presence of a wave field, including the wave contribution to the stress-energy and spin angular momentum tensors.