Effect on Synchrotron Motion by Modulation of High Frequency Voltage

In the classical mechanics frame and small amplitude approximation,the longitudinal motion equation of the particle was reduced to the Mathieu equation based on the perturbed Hamiltonian.It was pointed out that there is a series of new resonance lines in quasi-isochronous synchrotron by the perturbation.The boundary curve and the stop width for order instability zone one were derived by the method of perturbabtion.Also it was pointed that the system is at the critical point when the phase becomes equal to the phase of the instable point for the synchrotron after it crossed the stop band,and the critical condition of the high frequency voltage modulation was derived.