A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917103900033024 |
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| author | Colombo, Leonardo |
| author_facet | Colombo, Leonardo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20538 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System Colombo, Leonardo Mathematical Physics Differential Geometry Dynamical Systems Symplectic Geometry 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. |
| title | A Presymplectic and Symmetry Reduced Formulation of the Maxwell Vlasov System |
| topic | Mathematical Physics Differential Geometry Dynamical Systems Symplectic Geometry |
| url | https://arxiv.org/abs/2511.20538 |