Tune shift with amplitude 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. In classical mechanics, a simple example of a system with tune shift with amplitude is a pendulum. In accelerator physics, both the transverse and the longitudinal dynamics show tune shift with amplitude. A simple model of the transverse dynamics is of an oscillator with a single sextupole, it is referred to as the Hénon map. Another model for this case is the Standard Map. An important example is the typical case of distributed
Tune shift with amplitude 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 formula_1 and the action formula_2. The tune shift with amplitude is then given by formula_3. The transformation to action-angle variables out of which the tune may be derived may be considered as a transformation to normal form. 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