Algebra of Invariants for the Vlasov-Maxwell System

TL;DR

The paper addresses the problem of characterizing invariants for the relativistic and nonrelativistic multispecies Vlasov-Maxwell system and clarifies how the constraints $\nabla\cdot \mathbf{B}=0$ and $\nabla\cdot \mathbf{E}=\rho$ shape the invariant algebra. It develops invariant densities and fluxes directly from the noncanonical Hamiltonian structure and analyzes the Poisson algebra of invariants, showing that on the Casimir leaf the relativistic invariants realize the Poincaré algebra while the nonrelativistic ones realize the Euclidean algebra. Key contributions include explicit expressions for Casimir fluxes, conserved quantities (momentum $\mathbf{P}$, energy $H$, center of mass $\mathbf{M}$, angular momentum $\mathbf{L}$), and a detailed Lie-algebra realization of invariants; the work also emphasizes the asymmetric roles of $\nabla\cdot \mathbf{B}=0$ and $\nabla\cdot \mathbf{E}-\rho=0$ in sustaining a Hamiltonian structure. The findings have practical significance for structure-preserving numerical schemes (e.g., Dirac constraint methods) and deepen the understanding of symmetry, conservation, and constraint handling in Vlasov-Maxwell dynamics.

Abstract

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, $\nabla\cdot \mathbf{B}$.