By Etienne Forest

This ebook illustrates a thought compatible to monitoring codes, which the writer has built through the years. monitoring codes now play a primary function within the layout and operation of particle accelerators. the idea is totally defined step-by-step with equations and real codes that the reader can bring together and run with freely on hand compilers.

In this e-book, the writer pursues an in depth technique in accordance with finite “s”-maps, for the reason that this can be extra normal so long as monitoring codes stay on the centre of accelerator layout. The hierarchical nature of software program imposes a hierarchy that places map-based perturbation thought above the other tools. The map-based process, maybe sarcastically, permits eventually an implementation of the Deprit-Guignard-Schoch algorithms extra devoted than whatever present in the normal literature. This hierarchy of tools isn't a private selection: it follows logically from monitoring codes overloaded with a truncated energy sequence algebra package.

After defining abstractly and in brief what a monitoring code is, the writer illustrates many of the accelerator perturbation conception utilizing a precise code: PTC. This e-book could appear like a guide for PTC; although, the reader is inspired to discover different instruments in addition. The presence of an exact code guarantees that readers can have a device with which they could attempt their figuring out. Codes and examples might be on hand from quite a few websites considering the fact that PTC is in MAD-X (CERN) and BMAD (Cornell).

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Additional resources for From Tracking Code to Analysis: Generalised Courant-Snyder Theory for Any Accelerator Model

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The coefficients of cos μi and sin μi , in the de Moivre parametrisation of the one-turn matrix in Eq. 28), i = 1, n in n-d-f. See Sect. 2 for n ≥ 2. The Definitions 1 and 2 are general and extend to nonlinear coupled systems. Definition 3 extends to linear coupled lattice functions. In the coupled case, the matrix multiplying cos μi is not the identity and contains useful information. In fact, in the non-symplectic case, Definition 3 is the only sensible definition of the lattice functions since there are no invariants and no averages but there is a de Moivre representation as can be seen in Sect.

On a linear map, we proceed by first looking at the action of M on a linear function without any constant part. I will now show that the matrix for M in the space of linear functions is the transpose of the usual matrix M. 4 Such a function can be written as: f (z) = v1 z 1 + v2 z 2 . 37) The map M acts on f in the usual way of a map transforming functions: v1 M1i z i + v2 M2i z i (M f ) (z) = f (M z) = i = M †v 1 z1 + M †v 2 z2 . 38) is also the “pull back” operator of differential forms; not surprisingly the invariance of the Poisson under M is equivalent to the invariance of the canonical two-form i=1,N dz 2i−1 ∧ dz 2i under the pull back M .

7), we can rewrite Eq. 6): N F ◦ a ◦ a −1 + F ◦ a ◦ r ◦ a −1 + · · · + F ◦ a ◦ r ◦ a −1 N F ◦ a + F ◦ a ◦ r + ··· + F ◦ a ◦ rN = ◦ a −1 N →∞ N F + F ◦ r + ··· + F ◦ rN ◦ a −1 N → ∞ . 9) 1 By “general case”, I really mean general: 20 degrees of freedom, sixth-order, with magnet modula- tion and parameter dependence, spin, etc…!!! But concentrate on the simplest case always bearing in mind the feasibility of the general case. 2 Why Phasors and Normal Forms? 9) expresses a simple fact: the average of a function F is gotten from the average of the transformed function F = F ◦ a under a rotation.

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