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In recent years, several gauge-symmetric particle-in-cell (PIC) methods havebeen developed whose simulations of particles and electromagnetic fieldsexactly conserve charge. While it is rightly observed that these methods' gaugesymmetry gives rise to their charge conservation, this causal relationship hasgenerally been asserted via ad hoc derivations of the associated conservationlaws. In this work, we develop a comprehensive theoretical grounding for chargeconservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. ForLagrangian variational PIC methods, we apply Noether's second theorem todemonstrate that gauge symmetry gives rise to a local charge conservation lawas an off-shell identity. For Hamiltonian splitting methods, we show that themomentum map establishes their charge conservation laws. We define a new classof algorithms -- gauge-compatible splitting methods -- that exactly preservethe momentum map associated with a Hamiltonian system's gauge symmetry -- evenafter time discretization. This class of algorithms affords splitting schemes adecided advantage over alternative Hamiltonian integrators. We apply thisgeneral technique to design a novel, explicit, symplectic, gauge-compatiblesplitting PIC method, whose momentum map yields an exact local chargeconservation law. Our study clarifies the appropriate initial conditions forsuch schemes and examines their symplectic reduction.

The geometric theory of charge conservation in particle-in-cell simulations