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.