Beltrami磁场的范式与近轴展开
Normal Forms and Near-Axis Expansions for Beltrami Magnetic Fields
AI总结：
针对磁轴附近的环面真空与Beltrami磁场，利用Bishop坐标、Hamilton结构与Floquet理论建立Birkhoff-Gustavson范式，并给出至四阶的良定义近轴展开。
摘要：
本文针对磁轴邻域内的环面磁场构型，得到了一种变换至Birkhoff-Gustavson范式的形式级数。研究采用Bishop的最小旋转坐标，在轴附近构建局部正交标架；即使曲率存在零点，该标架中的度量仍为对角形式。我们以统一方式处理真空场和无力（Beltrami）场情形，并指出向量势本质上就是Hamilton动力学中的Poincaré-Liouville一形式，而所得磁场对应于非自治单自由度系统的标准二形式。文中得到了正则坐标，并利用Floquet理论变换到一个使最低阶Hamilton量成为自治的标架。所得磁轴可以是椭圆型或双曲型，并处理了共振椭圆情形。结果表明，所得场展开在所有阶上均是良定义的，并被显式计算至四阶。文中还给出了一个靠近1:3共振、具有常挠率轴的例子。
英文摘要：
A formal series transformation to Birkhoff-Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation-minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of vacuum and force-free (Beltrami) fields in a unified way, noting that the vector potential is essentially the Poincaré-Liouville one-form of Hamiltonian dynamics, and the resulting magnetic field corresponds to the canonical two-form of a nonautonomous one-degree-of-freedom system. Canonical coordinates are obtained and Floquet theory is used to transform to a frame in which the lowest-order Hamiltonian is autonomous. The resulting magnetic axis can be elliptic or hyperbolic, and resonant elliptic cases are treated. The resulting expansion for the field is shown to be well-defined to all orders, and is explicitly computed to degree four. An example is given for an axis with constant torsion near a 1:3 resonance.
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。
