ANALYTIC SOLUTION OF PARTICLE MOTION IN A DOUBLE RF SYSTEM

Analytic solutions for particle motion in a double rf system having a harmonic ratio of two are obtained. The effects of time dependent rf p hase and voltage modulation on synchrotron motion are studied. Paramet ric resonances arising from a weak perturbation are analyzed in terms of the action-angle variables of the unperturbed Hamiltonian. Sum rule s for the strength functions are derived. We find that the tree of bif urcation branches for these parametric resonances follows the characte ristic tune of the unperturbed Hamiltonian. The basins of attraction f or the dissipative double rf system are also studied in numerical simu lations.