On the Differential Algebra Underlying the COSY INFINITY Computer Code Due to M. Berz Page: 2 of 29
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On the Differential Algebra Underlying the
COSY INFINITY Computer Code Due to M. Berz
Accelerator Development Department
Brookhaven National Laboratory
Upton, NY 11973, USA
The mathematical foundations of the differential algebraic approach
to beam optics, due to M. Berz, are described. They are simplified by
identifying the underlying algebraic structure with the well known algebra
of truncated polynomials. Concrete examples of derivations in this algebra,
consistent with the truncation operation, are given.
There are effective methods for solving systems of differential equations, to any order
in z, 1-4
z' = F (z, 6), (S - parameters)
z (si) = zi.
Here z can be a multidimensional vector like, that used in particle beam optics, for example.
The solution of this problem can be stated as a mapping between the initial variables
zi = z (si), and the final ones zf = z (Sf)
Zf = M (zi, 6), (1.2)
The map M is of particular interest for accelerator physics as it contains important in-
formation about various characteristics of a given ring, (nonlinearities, dependence on
external parameters 6), and can be used for fast tracking over many turns. The Taylor
expansion coefficients of the map, the derivates, more exactly
akz f _ kM
az' az , k= 1,2, ...,n, (1.3)
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V., Garczynski. On the Differential Algebra Underlying the COSY INFINITY Computer Code Due to M. Berz, report, July 1, 1992; United States. (digital.library.unt.edu/ark:/67531/metadc867275/m1/2/: accessed September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.