INVESTIGATION OF PHASE TRAJECTORIES OF PARTICLE MOTION IN A SYNCHROTRON NEAR THE NONLINEAR RESONANCE OF THIRD ORDER

The nonlinear resonance of third order plays an important role in the particle dynamics in circular accelerators, colliders and storage rings and is widely used for slow extraction of particles from synchrotrons. Consideration is carried out in the canonical variables X, Y which at a given accelerator azimuth are simply related to the angle and the deviation of the circulating particles relative to the equilibrium orbit. The problem is reduced to the construction of the phase trajectories, which are the curves of the third or fourth order and determine the type of the motion near the resonance under consideration. The construction of the phase trajectories is performed by the Klein’s perturbation method. The influence on the particles dynamics of octupole component of the magnetic fields is investigated.