This paper gives a first application of the reduced-phase-space Lagrangian for kinetic theories obtained in a sister paper in this issue by means of Kruskal's averaging coordinates. The approximations made within Kruskal's formalism are of first order in the smallness parameter $\epsilon$, given by the ratio of the gyroperiod to the macroscopic time scale, which is the same as the ratio of the gyroradius to the macroscopic scale length in the drift-kinetic case, or as the ratio of the amplitudes of the fluctuations to the background fields in the gyrokinetic case. This paper presents methods and results concerning local conservation laws for the density of gyrocentres and the charge, energy, momentum and angular momentum. A very important feature of our treatment is that throughout the theory is gauge invariant. The methods consist of a modified Noether formalism with gauge-invariant shift variations which in a very straightforward way lead to, in particular, the symmetric energy–momentum tensor instead of the canonical tensor. The shift variations are defined both within the reduced phase space, which does not contain the gyroangle, and also for gyroangle-dependent quantities which subsequently have to be averaged. A clear definition of the Lagrange density needed for the derivation of the local conservation laws for energy, momentum and angular momentum is given. The discovery of combinations of terms such as the polarization and the magnetization allows the conservation laws to be cast in a very clear form affording insight into their structure.
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