New Formulation of the Equation for Synchrotron Oscillations

We develop an orbit theory for synchrotron oscillations using the orbit length, s, as an independent variable. This is commonplace for static magnetic fields (storage rings). We extend this to the case of adiabatically varying magnetic fields (synchrotrons). Contrary to conventional treatments, betatron acceleration terms appear in both the energy and phase equations. We derive one-turn difference equations in the linear and adiabatic approximations. By a smooth approximation instead of the traveling-wave approximation, and by combining the two equations, we obtain a differential equation where the betatron acceleration terms are canceled. This equation is an extension of McMillan’s equation to the case of strong-focusing synchrotrons.