Abstract
A normal form is an idealisation of the nonlinear world based on an extrapolation from the linear domain. This is its central limitation. But if we accept it, with reservations, it entails that it can only be understood if the linear system has first been studied. In this chapter, I examine several linear systems upon which a nonlinear normal form can be defined: the full harmonic sink, the Jordan coasting beam, the radiative stochastic beam envelope and the periodically modulated magnets. Only spin is left out and treated later. In the case of the damped harmonic coupled oscillator, I introduce its de Moivre representation and some consequences.
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Notes
- 1.
Actually the results of this section apply to N degrees of freedom; but in accelerators \(N=3\) is the maximum size of the symplectic space.
- 2.
The dispersive \(\zeta \) of the Japanese trio, Ohmi, Hirata and Oide, cannot be deduced from a cavity-less system because nothing depends on the time of arrival in the absence of longitudinal(temporal) focussing. Thus the \(\zeta \) functions can only emerge from an examination of the full three-dimensional oscillator.
- 3.
Sometimes called Napoleon in the USA.
- 4.
If you do so, make sure for the sake of Chap. 8 that you keep a copy of the original lattice, markers included.
- 5.
Nonlinear components of the phase slip may enter as well.
- 6.
The paper of Tomàs is not useless, I am obviously exaggerating. The code is still controlled by users who desire once in a while some analytic understanding! In fact in Sect. 5.3 I perform a calculation which parallels Tomàs’ calculation apparently guilty myself of “uselessness.”
- 7.
Translation, like canonical and propaganda, is also a word we owe to the Church. For example, one talks of the “translation” of the relics of St-Nicholas from Myra to Bari!
- 8.
Contigit hoc nullo nisi te sub principe, Caesar: cum duo pugnarent, uictor uterque fuit. From Martial’s Liber de Spectaculis.
- 9.
36 fluctuations but only 21 independent ones because \(\left\langle {{z}_{i}{z}_{j}}\right\rangle = \left\langle {{z}_{j}{z}_{i}}\right\rangle \).
References
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Forest, E. (2016). Classification of Linear Normal Forms. In: From Tracking Code to Analysis. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55803-3_4
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DOI: https://doi.org/10.1007/978-4-431-55803-3_4
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