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A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System

Leonardo Colombo

TL;DR

This work develops a comprehensive geometric formulation of the Maxwell–Vlasov system using the infinite-dimensional Skinner–Rusk presymplectic framework, with the Gotay–Nester–Hinds constraint algorithm recovering the full dynamics as a hierarchy of compatibility conditions. Through reduction by the particle-relabelling group Diff(TR^3), it unifies Euler–Poincaré and Marsden–Weinstein–Morrison–Greene Hamiltonian structures, and shows how these persist under equilibria with partial symmetry breaking. The paper further introduces energy–Casimir stability analysis on the reduced presymplectic space, and extends the framework to external antenna control via affine Hamiltonian controls, yielding controlled Lie–Poisson dynamics and Casimir shaping to stabilize selected equilibria. Collectively, the results provide a single geometric foundation connecting Lagrangian, Hamiltonian, gauge, reduction, and control perspectives for Maxwell–Vlasov plasmas, with clear implications for stability analysis and RF-plasma 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.

A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System

TL;DR

This work develops a comprehensive geometric formulation of the Maxwell–Vlasov system using the infinite-dimensional Skinner–Rusk presymplectic framework, with the Gotay–Nester–Hinds constraint algorithm recovering the full dynamics as a hierarchy of compatibility conditions. Through reduction by the particle-relabelling group Diff(TR^3), it unifies Euler–Poincaré and Marsden–Weinstein–Morrison–Greene Hamiltonian structures, and shows how these persist under equilibria with partial symmetry breaking. The paper further introduces energy–Casimir stability analysis on the reduced presymplectic space, and extends the framework to external antenna control via affine Hamiltonian controls, yielding controlled Lie–Poisson dynamics and Casimir shaping to stabilize selected equilibria. Collectively, the results provide a single geometric foundation connecting Lagrangian, Hamiltonian, gauge, reduction, and control perspectives for Maxwell–Vlasov plasmas, with clear implications for stability analysis and RF-plasma 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.

Paper Structure

This paper contains 20 sections, 24 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Infinite-Dimensional Skinner--Rusk Formalism
  3. Infinite-dimensional configuration spaces
  4. The Gotay--Nester--Hinds Constraint Algorithm
  5. Lagrangian Formulation of Maxwell--Vlasov in the SR Framework
  6. Reduction of the Skinner--Rusk Maxwell--Vlasov System
  7. The Reduced Equations: Lagrangian picture
  8. Hamiltonian Picture and Lie--Poisson Brackets
  9. Step 1: Skinner--Rusk reduction and the semidirect product structure.
  10. Step 2: GNH algorithm and the constraint manifold $C_\infty$.
  11. Step 3: Identification with the Marsden--Weinstein Lie--Poisson structure.
  12. Step 4: Passage to Eulerian coordinates and the Morrison--Greene bracket.
  13. Symmetry Breaking and Reduced Dynamics Near Equilibria
  14. Energy--Casimir Methods and Stability in the Reduced SR Framework
  15. Controlled Symmetry Breaking and Hamiltonian Control in the Skinner--Rusk Maxwell--Vlasov Framework
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $Q$ be an infinite-dimensional configuration manifold whose cotangent bundle $T^\ast Q$ carries the canonical symplectic form $\omega_{\mathrm{can}}$. Then the two-form on $W=TQ\oplus T^\ast Q$ is closed and weakly non-degenerate. In particular, $(W,\Omega)$ is a presymplectic manifold.

Figures (1)

  • Figure 1: Conceptual diagram: antenna control vector fields $X_{B_a}$ act on the SR manifold before and after GNH reduction, modifying symmetry directions and Casimir leaves.

Theorems & Definitions (69)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Definition 3.1
  • Proposition 3.2
  • ...and 59 more