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Therefore, it is virtually impossible to obtain a 10-million-turn (the lifetime of an SSC injection lattice) survival plot for the SSC with such an element-by-element tracking approach. 2 One-Turn-Map Trackings As discussed in the previous section, a conventional element-by-element tracking method is virtually impossible for the desired lifetime (10 million turns) in the SSC injection lattice. Mapping techniques are currently among alternatives under consideration for achieving such a task. In this section, we shall first discuss how one-turn maps are constructed.
Step 4. Re-expanded Taylor Map Tracking. Use the two re-expanded Taylor maps for long-term tracking up to the expected turn. Compare the survival 48 plots. If the dynamic apertures from the two re-expanded Taylor map trackings match reasonably well, then we have obtained the desired results. If the dynamic apertures do not agree within a reasonable tolerance, then there are two possibilities to consider. One possibility is that the Taylor map extracted in Step 1 does not carry enough accurate information for the lattice.
Further studies show that mu even lower-order differential algebraic Taylor map, after improvement of symplecticity through re-expansion with its Lie transformations, can also give a reasonably good measurement of the dynamic aperture of the lattice. 2: For a differential algebraic Taylor map directly extracted from a symplectic tracking program, the degree of accuracy of carrying the lattice information is higher than the degree of mathematical symplecticity. The high orders in the map are usually kept not for important lattice information but to provide the required symplecticity.
Appl of Differential Algebra to Single-Particle Dynamics in Storage Rings