Tune shift with amplitude
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The tune shift with amplitude is an important concept in circular accelerators or synchrotrons. The machine may be described via a symplectic one turn map at each position, which may be thought of as the Poincaire section of the dynamics. A simple harmonic oscillator has a constant tune for all initial positions in phase space. Adding some non-linearity results in a variation of the tune with amplitude. Amplitude may refer to either the initial position, or more formally, the initial action of the particle.
Definition
Consider dynamics in phase space. These dynamics are assumed to be determined by a Hamiltonian, or a symplectic map. For each initial position, we follow the particle as it traces out its orbit. After transformation into action-angle coordinates, one compute the tune and the Action (physics) . The tune shift with amplitude is then given by
Significance
The tune shift with amplitude is important as a measure of non-linearity of a system. A linear system will have no tune shift with amplitude. Further, it can be important regarding the stability of the system. When the tune reaches resonant values, it can be unstable, and thus a tune-shift with amplitude can limit the stability region, or dynamic aperture.
Examples
An important example is that involving distributed sextupoles. In the case of a single sextupole, it is referred to as the Henon map. Another model for this case is the Standard Map.