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Lth order. This requires only a single calling statement of the corresponding Z:ib subroutine, using much less computer time than does the one-turn Taylor map extraction of the SSC. Step 3. Higher-Order Taylor Map Expansions Depending on the expected turns of tracking, the Lie transformations are converted back to two higher-order Taylor maps up to the same order. For example, the 14th-order Taylor maps would be considered for the 107-turn (lifetime) tracking of the SSC injection lattice. The two re-expanded higher-order Taylor maps differ in that one is converted from the series of homogeneous Lie transformations up to the highest order in the series, while the other is converted from up to the second highest order in the series.

Once a one-turn, high-order Taylor map is obtained, it can be directly used for particle trackings, but with care. It can also be symplectified by conversion into a series of homogeneous Lie transformations (a Dragt-Finn factorization map). These Lie transformations can be transferred back to even higher-order Taylor maps for long-term trackings. They can also be converted into normal forms for various analyses and into kick factorizations for symplectic trackings. The series of homogeneous Lie transformations also can be combined into a single, non-homogeneous Lie transformation through use of the CBH theorem.

18). _mn o transformation of order i, each factorization subset should be denoted as _ , and the required number of bases in the subset is r mnĀ° = (m + 1)(n + 1)(o+ 1), where m + n + 0 = i. Therefore, the minimum required kick number for the 6-dimensional case would be - 3 3 /' where f_ is the order of the map in terms of the Taylor expansion, and ](i/j) means integer operation for the division i/j. For example, Fmin = 80 if f_ = 9 and Fmin "- 125 if S2 = 11. 3 Order-by-Order Kick Factorizations To kick-factorize the Dragt-Finn factorization map given by Eq.

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Appl of Differential Algebra to Single-Particle Dynamics in Storage Rings


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