解耦的几何方法
A Geometrical Method of Decoupling
摘要：
调谐和匹配光束分布的计算是分析圆形加速器的基本步骤。如果存在某些对称性--如中平面对称性--那么就有可能使用著名的柯朗-斯奈德理论分别处理电子感应加速器在水平、垂直平面和（在某些情况下）纵向运动中的运动，或者应用前面已经描述过的变换，例如滕和爱德华兹的方法。在前面的文章中，它已表明，这种方法需要修改的处理等熵回旋加速器与不可忽略的空间电荷力。不幸的是，这种修改在数值上并不像预期的那样稳定，而且仍然不清楚这种扩展是否适用于所有的情况。因此，系统地推导出一种更一般的处理方法似乎是可取的。在第二篇论文中，作者建议使用真正的狄拉克矩阵作为基本工具，耦合线性光学，并给出了一个简单的配方解耦正定哈密顿与虚特征值。本文对这种方法进行了推广和简化，给出了一种求解特征值在实轴和虚轴上的Hamilton矩阵解耦的简单方法。它示出，这种代数解耦是密切相关的几何“解耦”的正交化的矢量$\vec E$，$\vec B$和$\vec P$，被引入了所谓的“机电等效”。我们提出了一个保结构的块对角化的辛或哈密顿矩阵，分别。当迭代使用时，解耦算法也可以应用于n维系统，并且需要${\cal O}(n^2)$迭代以收敛到给定精度。
英文摘要：
The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries - like midplane symmetrie - are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane and (under certain circumstances) the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as for instance the method of Teng and Edwards. In a preceeding paper it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all thinkable cases. Hence a systematic derivation of a more general treatment seemed advisable. In a second paper the author suggested the use of real Dirac matrices as basic tools to coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. It is shown that this algebraic decoupling is closely related to a geometric "decoupling" by the orthogonalization of the vectors $\vec E$, $\vec B$ and $\vec P$, that were introduced with the so-called "electromechanical equivalence". We present a structure-preserving block-diagonalization of symplectic or Hamiltonian matrices, respectively. When used iteratively, the decoupling algorithm can also be applied to n-dimensional systems and requires ${\cal O}(n^2)$ iterations to converge to a given precision.
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