A Geometrical Method of Decoupling

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.