磁场作为1½维哈密顿系统的全局实现
Global realisation of magnetic fields as 1$\frac{1}{2}$D Hamiltonian systems
AI总结：
本文研究三维流形上无散向量场的场线动力学何时可全局实现为非自治1½维哈密顿系统，通过Moser技巧和Poincaré截面平面性条件建立识别条件，并利用纤维丛单值分类证明当截面为圆盘或圆环时定义域必为实心或空心环面，从而场线动力学总能实现为哈密顿系统。
摘要：
本文回顾了n+½维非自治哈密顿系统的概念，将其动力学描绘为与余辛流形上最大秩闭二形式相关的Reeb场的流，并自然地分解为类时分量和哈密顿分量。随后，本文研究了在何种条件下，一个微分同胚于圆周上平凡纤维丛的连通紧致三维流形（可能带边界）上的（切向）无散向量场的场线动力学能够反过来被识别为非自治1½维哈密顿系统。在场横截于一个全局紧致Poincaré截面的假设下，对Moser技巧的改编表明所有这样的场都是局部哈密顿的。在进一步假设Poincaré截面为平面时，建立了完全的识别，这一假设关键地意味着（连同Dirichlet边界条件）每个截面上生成闭一形式的同调类是常值。通过回顾利用单值表示对圆周上纤维丛的分类，本文指出，一旦所讨论的场的Poincaré截面微分同胚于圆盘或圆环，其定义域必然微分同胚于实心或空心环面，因此其场线动力学总是可以被识别为非自治哈密顿系统。
英文摘要：
The paper reviews the notion of $n+\frac{1}{2}$D non-autonomous Hamiltonian systems, portraying their dynamics as the flow of the Reeb field related to a closed two-form of maximal rank on a cosymplectic manifold, and naturally decomposing into time-like and Hamiltonian components. The paper then investigates the conditions under which the field-line dynamics of a (tangential) divergence-free vector field on a connected compact three-manifold (possibly with boundary) diffeomorphic to a trivial fibre bundle over the circle can be conversely identified as a non-autonomous $1\frac{1}{2}$D Hamiltonian system. Under the assumption that the field is transverse to a global compact Poincaré section, an adaptation of Moser's trick shows that all such fields are locally-Hamiltonian. A full identification is established upon further assuming that the Poincaré sections are planar, which crucially implies (together with Dirichlet boundary conditions) that the cohomology class of the generating closed one-forms on each section is constant. By reviewing the classification of fibre bundles over the circle using the monodromy representation, it is remarked that as soon as the Poincaré section of an alleged field is diffeomorphic to a disk or an annulus, the domain is necessarily diffeomorphic to a solid or hollow torus, and thus its field-line dynamics can always be identified as a non-autonomous Hamiltonian system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
