Abstract
I describe in some details how a nonlinear normal form is implemented in a library like my own FPP. I also discuss the first departure from the harmonic normal form: the one-resonance normal form. A very simple magnet modulation is treated theoretically. I leave accelerator physics with the introduction of a map which displays a limit cycle which breaks into what looks like a strange attractor.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
I remind the reader who skipped too many sections of this book that it is precisely the map that acts on functions by substitution that can be represented by Lie operators. In the linear case, one is often content with the usual matrix acting on rays. The Lie operators are not that useful. The one-turn map is the critical exception: the Courant-Snyder invariant is the Lie operator!
- 2.
Any sextupolar system requires at least a second-order calculation of the tune shifts. They determine the existence of islands.
- 3.
The reader will find a description of this resonance at the lowest order in a paper by Schmidt and Franchetti in Ref. [9].
- 4.
In the case of a couple resonance, \(\mathbf {a}\cdot \mathbf {J}\) of Eq. (5.44) is an invariant by virtue of being orthogonal to the plane of the resonance.
- 5.
Here by “un-normalised” I mean “normalised by a linear map” only.
- 6.
\({\nu }_x\), \(2{\nu }_x\) and \(3{\nu }_x\) driving terms are cleansed to second-order in sextupole strength.
- 7.
I need to be very explicit here because Lee’s book [10] is famous, especially in Asia where I work, and the results presented here seem to contradict his pronouncements. Since his pronouncements are couched in the usual East Asian vagueness, I felt the need to clarify the origin of the apparent discrepancy.
- 8.
I am not a specialist in electronics, but limit cycles could be perhaps found in the electronic circuitry of an accelerator. But that is pushing things a bit far...Look at Ref. [11] for example.
- 9.
Consult supplemental Chap. 10 to see that one must choose the phasors judiciously to insure this simple interpretation beyond linear matrices.
References
F. Willeke, F. Schmidt, Proceedings of the First European Particle Accelerator Conference (Joint Accelerator Conferences Website, CERN, 1988), p. 908
Y. Wu, E. Forest, KEK Preprint 2005–108 A (2006)
A.J. Dragt, Lectures on Nonlinear Orbit Dynamics, vol. 87, p. 147 (American Institute of Physics, New York, 1981)
E. Forest, Beam Dynamics: A New Attitude and Framework (Harwood Academic Publishers, Amsterdam, 1997)
K. Makino, M. Berz, Nucl. Instr. Meth. A558, 346 (2005)
A. Bazzanni, P. Mazzanti, G. Servizi, G. Turchetti, Nuovo Cimento B 51 (1988)
A. Bazzani, E. Todesco, G. Turchetti, G. Servizi, Physica D 64 (1993)
E. Forest, J. Irwin, in Nonlinear Problems in Future Particle Accelerators, Capri, Italy, ed. by W. Scandale, G. Turchetti (World Scientific, Singapore, 1990), p. 46
F. Schmidt, G. Franchetti, Phys. Rev. Lett. 114 (2015)
S.Y. Lee, Accelerator Physics (World Scientific Publishing, Singapore, 2004)
E. Tamas̆evic̆i\(\bar{\rm u}\)tė et al., Nonlinear Analysis: Modell. Control 13(2), 241 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer Japan
About this chapter
Cite this chapter
Forest, E. (2016). Nonlinear Normal Forms. In: From Tracking Code to Analysis. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55803-3_5
Download citation
DOI: https://doi.org/10.1007/978-4-431-55803-3_5
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55802-6
Online ISBN: 978-4-431-55803-3
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)