Using smooth approximation for beam dynamics investigation in superconducting linac

The superconducting (SC) linac consists of some different classes of the identical cavities. The each cavity based on a superconducting structure with a high accelerating gradient. The distance between the cavities is equal to acceleration structure period L. By specific phasing of the RF cavities one can provide a stable particle motion in the whole accelerator. Three dimensional equation of motion for ion beam in the Hamiltonian form is derived in the smooth approximation for superconducting linac. The nonlinear ion beam dynamics is investigated in such accelerated structure.