Abstract
I extend in this chapter the concept of a normal form to nonlinear systems in 1-d-f. I try to contrast Hamiltonian methods and one-turn map methods. The pendulum is solved by “undergraduate tools” and the calculation is corroborated by the perturbation tools of this book. The standard map, which is just the map for an RF-cavity restricted to the longitudinal plane, is introduced as the archetype of the trouble maker: it can display resonances and even chaos if “pushed” too hard.
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Notes
- 1.
It is possible to design pure nonlinear systems which are stable but this is not commonly considered in particle accelerators.
- 2.
To my knowledge, there is only one Hamiltonian taken out of a hat which was surprisingly found to be integrable despite having no apparent symmetries: the Toda lattice.
- 3.
Cosy-Infinity is the ultimate matrix code equipped with TPSA tools beyond the scope of our discussion, interval arithmetic for example.
- 4.
FPP can be used alone. But instead of introducing more mysterious calls, I initialised PTC in 1-d-f which then ripples down to FPP. As I said, I never use the tracking code PTC in this pendulum program.
- 5.
I have routines that go back and forth between Hamiltonians and vector fields, but here I wanted to be explicit for this simple example.
- 6.
Notice the order which is reversed from the usual Taylor map: remember the transpose matrix \({M}^{\dagger }\) of Sect. 2.3.1!
- 7.
Well...four different ways but I also include the call to FPP which is actually a generalised version of the application of vector fields to one-turn maps.
- 8.
FPP has a routine called flatten_c_factored_lie that performs this task, but here I prefer to be explicit.
- 9.
If the map is not symplectic due to small computation errors, it pays to choose a path that avoids the region of maximum violation. For example, in the days of the defunct Super Conducting Super Collider, a member of SSC asked me to “symplectify” a map of the Jordan normal form type (see Sect. 4.2) that violated the symplectic condition mostly in the longitudinal direction due to truncation errors. I did not use the diagonal but avoided the longitudinal plane as much as possible in the integration path. This is also useful when a transverse map is known through measurements as a function of energy: one can reconstruct almost completely the longitudinal part of the map by avoiding it in the integration path.
- 10.
This is “s”-dependent Hamiltonian perturbation theory where H is Fourier-transformed in the time-like variable s often mapped into an angular variable \(\theta \) as I do in Chap. 8. It is commonly described in many accelerator books though rarely carried through even on simple lattices.
References
M. Berz, Technical report, Michigan State University (unpublished)
M. Berz, Part. Accel. 24, 109 (1989)
A. Chao, Technical Report No. SLAC-PUB-9574, Stanford Linear Accelerator (unpublished)
G. Guignard, Technical Report No. CERN 78–11, CERN (unpublished)
S.Y. Lee, Accelerator Physics (World Scientific Publishing, Singapore, 2004)
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Forest, E. (2016). The Nonlinear Transverse Normal Form: One Degree of Freedom. In: From Tracking Code to Analysis. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55803-3_3
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DOI: https://doi.org/10.1007/978-4-431-55803-3_3
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