Local conservation laws for the Maxwell-Vlasov and collisionless kinetic guiding-center theories.

Abstract

expressions for the and flux densities were determined. Here the full energy-momentum tensor and the angular momentum tensor are obtained via Noether's theorem. The appropriate symmetries of these tensors are shown and the local conservation laws are proven. These results are of importance for applications; e.g., knowledge of the total angular momentum combined with the can lead to an energy principle for linear stability analysis. The tensors are obtained in the usual way by first deter­ mining the general expression for an arbitrary variation of the total· Lagrangian density in normal position space x. We note, however, that there is a slight complication be­ cause the particle part of the Lagrangian is primarily defined on an extended space Y=(YY2) where y, is iden­ tical to x and Y2 is an additional coordinate that is needed in order to describe guiding centers. By means of transla­ tional invariance in x space and time, and rotational in­ variance in x space· the canonical tensors are obtained. These tensors are not gauge invariant, but each can be split into a divergence-free gauge-invariant part and a divergence-free non-gauge-invariant part. The gauge­ invariant energy-momentum tensor turns out to be sym­ metric in its spatial components. This is shown to follow from its relationship to the gauge-iI,lVariant part of the an­ gular momentum tensor. All of these expressions are also applicable to relativistic theories. Two applications are considered: first we treat the Maxwell-Vlasov equations and show that the gauge­ invariant parts of our tensors reduce to the usual well­ known expressions. Following this we treat the Maxwell­ kinetic guiding-center theory based on Littlejohn's guiding-center equations of motion.4