Hamiltonian formulation for synchrotron oscillations and Sacherer's integral equation

Sacherer's integral equation, which is the basis of many works on bunched-beam instabilities, is derived from basic principles. Differences in the form of the equation appearing in previous work are analyzed. This involves critical analysis of the equations of synchrotron motion. The concept of coupling impedance is explained in connection with the time-domain concept of the wake field. The nonlinear nature of synchrotron and betatron motions is included naturally in the theory by using action-angle variables. This will clarify the discussion of Landau damping. (author)