A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System

TL;DR

This paper presents a comprehensive geometric formulation of the Maxwell-Vlasov system using the Skinner-Rusk formalism, unifying particles, fields, and symmetries within a presymplectic framework, and extends to include equilibria and control mechanisms.

Contribution

It introduces a unified presymplectic geometric framework for Maxwell-Vlasov equations, incorporating reduction, symmetry breaking, and control within a single formalism.

Findings

Derivation of Maxwell-Vlasov equations from a presymplectic variational principle.

Natural emergence of constraints and gauge structure from geometry.

Extension to controlled symmetry breaking and plasma-antenna interactions.

Abstract

We develop a unified geometric formulation of the Maxwell-Vlasov system using the infinite-dimensional Skinner-Rusk (SR) formalism. In this framework, particles and fields are treated simultaneously within a single presymplectic manifold, and the Gotay-Nester-Hinds algorithm recovers the full Maxwell-Vlasov equations as the compatibility conditions of a single variational system. The hierarchy of constraints -- including Vlasov advection, Gauss and Faraday laws, and the electromagnetic gauge structure -- arises naturally from the presymplectic geometry of the SR formalism. Reduction by the diffeomorphism group of phase space produces a reduced presymplectic manifold whose dynamics reproduces both the Euler-Poincare formulation for the Vlasov sector and the Marsden-Weinstein/Morrison-Greene Lie-Poisson Hamiltonian structure. We further extend the construction to equilibria that partially break the relabeling symmetry, obtaining a translated SR system that yields an effective symplectic linearization, clarifies the appearance of Goldstone-type neutral modes, and provides a geometric foundation for the energy-Casimir method. Finally, we incorporate external antenna fields into this setting by introducing affine Hamiltonian controls and establishing a theory of controlled symmetry breaking within the reduced SR framework. This leads to controlled Lie-Poisson equations for plasma-antenna coupling and a geometric mechanism for stabilization through Casimir shaping and symmetry-selective forcing. The resulting picture provides a single presymplectic structure that unifies Lagrangian, Hamiltonian, gauge, reduction, and control-theoretic aspects of the Maxwell-Vlasov system.