标题： 正常形式和轴近似展开式对于贝特拉米磁场所述 显示英文标题

标题： Normal Forms and Near-Axis Expansions for Beltrami Magnetic Fields

摘要： 对环形磁场配置在磁轴附近的Birkhoff-Gustavson正规形式进行了一种形式级数变换。使用Bishop的最小旋转坐标，在轴附近获得了一个局部正交框架，即使曲率有零点，该框架中的度量也是对角的。我们以统一的方式处理了真空场和无体力场（Beltrami）的情况，注意到矢量势本质上是哈密顿动力学的Poincaré-Liouville一形式，所产生的磁场对应于非自治单自由度系统的典型二形式。获得了规范坐标，并使用Floquet理论转换到一个最低阶哈密顿量为自治的框架中。产生的磁轴可以是椭圆的或双曲线的，并处理了共振椭圆情况。显示了场的展开式在所有阶次上都是定义良好的，并显式计算到四阶。给出了一个在1:3共振附近具有恒定扭率的轴的例子。 显示英文摘要

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 Poincar\'e-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.