# Appl of Differential Algebra to Single-Particle Dynamics in

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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.

5). The horizontal axis represents the transverse amplitude of the particles, while the vertical axis represents the weighted distance D between the positions in the four-dimensional transverse phase space, obtained with the element-by-element tracking (Ztrack) and with the Taylor map tracking after one turn for each particle. {_ (/_x/_y)(yZ yM)2 +/3x_(pZ pM)2]z/2, where c3/3x/Ox = 0 = O_y/c3y at the measuring point. 2 × 10-s. 005 m in transverse amplitude, where the above figure shows the accuracy of the 1 lth-order map to be between 7 and 8 digits after one turn.

1 The Parameterized Courant-Snyder Matrix If we neglect the nonlinear multipole error effects, a one-turn respect to the dispersed closed orbit would be simply a parameterized Snyder map given by map with Courant- where M(6) is given by Eq. 3). The matrix Mo is symplectic, but M(6) generally is not unless it is expanded to the infinite order of 6. However, M(6) preserves one important symplectic property of the system, which is MT(6)SM(6) = S + a(6"+1), where n is the truncated order of M(_), and S is the symplectic identity discussed in Chapter 3.

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