Gauge Invariance and Conservation Laws in the Variational Formulation of Macro-Particle Plasma Models

Recently there has been significant interest in formulations of macro-particle models using variational methods. This is attractive because many of the inherent pathologies of traditional PIC methods are avoided. In reducing the Low Lagrangian to a grid for numerical computation it is generally the case that translational and gauge symmetries are lost, leading to the loss of both momentum and charge conservation. Using a macro-particle reduction of the charge distribution function we introduce a method to maintain these conservation laws in a gridded domain. By representing the vector and scalar potentials in different ways, we define charge and current densities which satisfy the discrete continuity equation. The continuity equation leads immediately to momentum conservation. If a convenient choice of representing the vector potential is made it is likely to lead to a corresponding scalar field representation which is unsatisfactory for computational use. Applying the Weyl gauge, however, can eliminate the need to ever perform these computations; knowing that a suitable basis for the scalar potential and thus a definition of the charge density exists is enough to maintain the conservation laws.