带源的Beltrami-Maxwell场的Liouville可积性：磁面的几何理论
Liouville integrability of Beltrami-Maxwell fields with sources: A geometric theory of magnetic surfaces
浏览论文内容
中文总结 AI 辅助
本文研究带源Maxwell场的几何理论，证明其可由Beltrami场构造，并在存在守恒量时化为辛流形上的Liouville可积系统，应用于London方程等。
中文摘要 AI 辅助
我们研究四维闵可夫斯基时空中具有非平凡源的经典Maxwell场。特别地，讨论了切触和辛几何方面。作为出发点，我们证明了几类带源的解可由所谓的Beltrami场构造，这些场是三维黎曼流形上的切触形式。此外，我们提供了这些类中的若干应用。其一是给出超导体London方程的一个解。另一个是，如果沿磁场向量场存在非平凡的守恒量，则Maxwell系统可视为四维辛流形上的Liouville可积Hamilton系统，该流形通过切触流形的辛化获得。作为例子，具有Lundquist模型、ABC流模型的一个受限情形等的Maxwell场在上述意义下被证明是可积的。
英文摘要
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.
