Liouville integrability of Beltrami-Maxwell fields with sources: A geometric theory of magnetic surfaces

Abstract

We study classical Maxwell fields with non-trivial sources in four-dimensional Minkowski spacetime. In particular, contact and symplectic geometric aspects are discussed. As a point of departure, it is shown that a few classes of solutions with sources are constructed from the so-called Beltrami fields, where these fields are contact forms on three-dimensional Riemannian manifolds. In addition, several applications in these classes are provided. One is to give a solution to the London equations for superconductors. Another one is that, if there is a non-trivial conserved quantity along the magnetic vector field, then the Maxwell system can be viewed as a Liouville integrable Hamiltonian system on a four-dimensional symplectic manifold, where this manifold is obtained by a symplectization of the contact manifold. As examples, Maxwell fields with the Lundquist model, a restricted case of the ABC flow model, and so on, are shown to be integrable in the above sense.

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BibTeX

@misc{goto2026liouville,
  title = {{Liouville integrability of Beltrami-Maxwell fields with sources: A geometric theory of magnetic surfaces}},
  author = {Shin-itiro Goto},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14242},
  eprint = {2609.14242},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14242},
}

RIS

TY  - UNPB
TI  - Liouville integrability of Beltrami-Maxwell fields with sources: A geometric theory of magnetic surfaces
AU  - Shin-itiro Goto
PY  - 2026
DA  - 2026/09/13
PB  - arXiv
DO  - 10.48550/arxiv.2609.14242
UR  - https://arxiv.org/abs/2609.14242
AB  - We study classical Maxwell fields with non-trivial sources in four-dimensional Minkowski spacetime. In particular, contact and symplectic geometric aspects are discussed. As a point of departure, it is shown that a few classes of solutions with sources are constructed from the so-called Beltrami fields, where these fields are contact forms on three-dimensional Riemannian manifolds. In addition, several applications in these classes are provided. One is to give a solution to the London equations for superconductors. Another one is that, if there is a non-trivial conserved quantity along the magnetic vector field, then the Maxwell system can be viewed as a Liouville integrable Hamiltonian system on a four-dimensional symplectic manifold, where this manifold is obtained by a symplectization of the contact manifold. As examples, Maxwell fields with the Lundquist model, a restricted case of the ABC flow model, and so on, are shown to be integrable in the above sense.
ER  -

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