Symplectic Orbit Theory of Synchrotron Oscillations

Abstract

We stress in this review paper that we should describe synchrotron oscillations using the length s of the central design orbit as an independent variable. A symplectic theory in this case is already available and we can incorporate important nonlinear features into the theory that are lost if we use the time t as the independent variable. We also settle the arguments about a closely related problem of the effects of betatron acceleration on synchrotron oscillations; the betatron acceleration affects not only the energy, but also the phase (arrival time) of the particles. When the equations for these two quantities are combined to form a second-order phase equation, the effects are cancelled in the adiabatic approximation.