Contact topology and electromagnetism: the Weinstein conjecture and Beltrami-Maxwell fields. (English) Zbl 07906069
Summary: 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.
©2024 American Institute of Physics
©2024 American Institute of Physics
MSC:
| 37J55 | Contact systems |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 53C40 | Global submanifolds |
| 53D10 | Contact manifolds (general theory) |
| 53Z05 | Applications of differential geometry to physics |
| 57K33 | Contact structures in 3 dimensions |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
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