We draw connections between contact topology and Maxwell fields in vacuo on three-dimensional closed Riemannian submanifolds in four-dimensional Lorentzian manifolds. This is accomplished by showing that contact topological methods can be applied to reveal topological features of a class of solutions to Maxwell’s equations. This class of Maxwell fields is such that electric fields are parallel to magnetic fields. In addition these electromagnetic fields are composed of the so-called Beltrami fields. We employ several theorems resolving the Weinstein conjecture on closed manifolds with contact structures and stable Hamiltonian structures, where this conjecture refers to the existence of periodic orbits of the Reeb vector fields. Here a contact form is a special case of a stable Hamiltonian structure. After showing how to relate Reeb vector fields with electromagnetic 1-forms, we apply a theorem regarding contact manifolds and an improved theorem regarding stable Hamiltonian structures. Then a closed field line is shown to exist, where field lines are generated by Maxwell fields. In addition, electromagnetic energies are shown to be conserved along the Reeb vector fields.
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August 2024
Research Article|
August 29 2024
Contact topology and electromagnetism: The Weinstein conjecture and Beltrami-Maxwell fields
Shin-itiro Goto
Shin-itiro Goto
a)
(Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing)
Center for Mathematical Science and Artificial Intelligence, Chubu University
, 1200, Matsumoto-cho, Kasugai, Aichi 487-8501, Japan
a)Author to whom correspondence should be addressed: sgoto@isc.chubu.ac.jp
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a)Author to whom correspondence should be addressed: sgoto@isc.chubu.ac.jp
J. Math. Phys. 65, 082704 (2024)
Article history
Received:
February 06 2024
Accepted:
August 07 2024
Citation
Shin-itiro Goto; Contact topology and electromagnetism: The Weinstein conjecture and Beltrami-Maxwell fields. J. Math. Phys. 1 August 2024; 65 (8): 082704. https://doi.org/10.1063/5.0202751
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