# The linear parameters and the decoupling matrix for linearly coupled motion in 6 dimensional phase space. Informal report [electronic resource].

- Published:
- Washington, D.C. : United States. Dept. of Energy, 1995.

Oak Ridge, Tenn. : Distributed by the Office of Scientific and Technical Information, U.S. Dept. of Energy. - Physical Description:
- 19 pages : digital, PDF file
- Additional Creators:
- Brookhaven National Laboratory, United States. Department of Energy, and United States. Department of Energy. Office of Scientific and Technical Information

##### Access Online

- Restrictions on Access:
- Free-to-read Unrestricted online access
- Summary:
- 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 6 × 6 matrix, R. R will be called the decoupling matrix. It will be shown that of the 36 elements of the 6 × 6 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, β{sub i}, α{sub i} = 1, 3, for the uncoupled coordinates, to be computed from the one period transfer matrix. An alternative procedure for computing the linear parameters, the β{sub i}, α{sub i} i = 1, 3, 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 α{sub i} and β{sub i}, i = 1, 3, and the elements of the decoupling matrix R. The procedure presented here for studying coupled motion in 6-dimensional phase space can also be applied to coupled motion in 4-dimensional phase space, where it may be a useful alternative procedure to the procedure presented by Edwards and Teng. In particular, it gives a simpler programming procedure for computing the beta functions and the emittances for coupled motion in 4-dimensional phase space.
- Report Numbers:
- E 1.99:bnl--61552

E 1.99: ad/rhic--132

ad/rhic--132

bnl--61552 - Subject(s):
- Other Subject(s):
- Note:
- Published through SciTech Connect.

03/01/1995.

"bnl--61552"

" ad/rhic--132"

"DE95009042"

Parzen, G. - Funding Information:
- AC02-76CH00016

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