Beam Physics Seminar
The Courant-Snyder parametrization of one-dimensional linear betatron motion is generalized to two-dimensional coupled linear motion. To represent the 4x4 symplectic transfer matrix the following ten parameters were chosen: four beta-functions, four alpha-functions and two betatron phase advances which have a meaning similar to the Courant-Snyder parametrization. Such a parametrization works equally well for weak and strong coupling and can be useful for analysis of coupled betatron motion in circular accelerators as well as in transfer lines. Relations between the proposed parametrization and Edwards-Teng parametrization are discussed, thus adding clarity to the physical meaning of both representations. The proposed set of beta-functions is bound up to the eigen-vectors. Similarly, the transfer matrix, the bilinear form describing the phase space ellipsoid and the second order moments are related to the eigen-vectors. Corresponding equations can be useful in interpreting tracking results and experimental data. As an example, the developed theory is applied to ionization cooling of muom beams in axially symmetric channels.
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