Gauge invariance and conservation laws in the variational formulation of macro-particle plasma models

In using variational methods to describe electrodynamics conservation laws are ensured by symmetries in accordance with Noether. Discretizing the system for computation, however, breaks one or more of the symmetries relied upon for the conservation of energy, momentum, or charge. We begin by reconciling two variational approaches to describing electrodynamics; by reducing the phase space distribution function to a collection of macro-particles with a fixed spatial structure and definite momentum, we show that working with electromagnetic potentials in the Low Lagrangian is identical to using the noncanonical Vlasov–Maxwell Poisson bracket with a Hamiltonian describing electric and magnetic fields. Favoring the Lagrangian formulation, we can show that while charge and momentum conservation are often linked they correspond to different symmetries which can be broken independently by the method of spatial discretization. Finally, we show that the machinery of variational methods naturally changes the potentials in a manner identical to a gauge transformation and that, if the discretized system is not gauge invariant, special care must be given to avoid contradictions in the resulting equations of motion.