Linear Parameters and the Decoupling Matrix for Linearly Coupled Motion in 6 Dimensional Phase Space

Details Linear Parameters and the Decoupling Matrix for Linearly Coupled Motion in 6 Dimensional Phase Space Linear Parameters and the Decoupling Matrix for Linearly Coupled Motion in 6 Dimensional Phase Space

1995-10-31

arxiv.org/abs/acc-phys/9510006v1

Physics

Accelerator Physics

18 pages, gzipped postscript paper (125k)

Scientific paper

It will be shown that starting from a coordinate system where the 6 phase space coordinates are linearly coupled, one can go to a new coordinate system where the motion is uncoupled by means of a linear transformation. The original coupled coordinates and the new uncoupled coordinates are related by a 6x6 matrix, R. R will be called the decoupling matrix. It will be shown that of the 36 elements of the 6x6 decoupling matrix R, only 12 elements are independent. This may be contrasted with the results for motion in 4-dimensional phase space, where R has 4 independent elements. A set of equations is given from which the 12 elements of R can be computed from the one period transfer matrix.This set of equations also allows the linear parameters for the uncoupled coordinates to be computed from the one period transfer matrix. An alternative procedure for computing the linear parameters and the 12 independent elements of the decoupling matrix R is also given which depends on computing the eigenvectors of the one period transfer matrix. These results can be used in a tracking program, where the one period transfer matrix can be computed by multiplying the transfer matrices of all the elements in a period, to compute the linear parameters and the elements of the decoupling matrix R.

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