Differential Algebraic Techniques

M. Berz, Michigan State U. The study of the behavior of motion in the vicinity of a chosen reference solution is a central problem arising in many sub-fields of dynamical systems, including beam dynamics. The Taylor expansions of these solutions can be obtained by solving the so-called variational equations, which in beam physics has been carried out to orders two and three in the code Transport[1, 2, 3], to orders three for example in the codes TRIO [4], GIOS, [5] and MaryLie [6], and to order five in the code COSY 5.0 [7]. This approach is very laborious in practice, and the development of the DA techniques described in the following has greatly simplified this endeavor in beam physics and other fields. In their latest versions [8, 9, 10, 11], the unprecedented accuracy these methods afford for the solution of differential equations has been recognized with the award of the R.E. Moore prize for rigorous computing. As used in our field [12], the DA techniques allow the convenient computation of high-order Taylor expansions of the transfer mapM which relates final particle coordinates ~zf to initial coordinates ~zi and parameters ~δ,

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