Algebra of Invariants for the Vlasov-Maxwell System

The algebra of invariants for both the relativistic and nonrelativistic multispecies Vlasov-Maxwell system is examined, including the case with a fixed ion background. Invariants and their associated fluxes are obtained directly from the Vlasov-Maxwell system. The invariants are shown to Poisson commute with the Hamiltonian and the rest of the Poisson bracket algebra of invariants is identified. Special attention is given to the role played by the monopole condition, $\mathrm{\nabla }\cdot \mathbf{B}$.