Canonical electrodynamics and continuous symmetries in discrete reductions

In reducing the Low Lagrangian to a finite system for numerical computation it is generally the case that basic physical properties such as momentum and charge conservation are lost. Using a macro-particle reduction of the phase space distribution function we explore the conservation properties of continuous and discrete systems; we see that charge and momentum conservation rely on properties of the continuous system that are effected differently by the method of discretization. We present here three discretization schemes which preserve, respectively, the symmetries needed to conserve neither charge nor momentum, only momentum but not charge, and both charge and momentum.