Envelope Hamiltonian of an intense charged-particle beam in periodic solenoidal fields

We introduce a Hamiltonian to describe the Kapchinskij-Vladimirskij envelope equation in the envelope phase space. The envelope Hamiltonian, in the presence of periodic focusing fields, can be decomposed into an unperturbed autonomous Hamiltonian and a time dependent perturbation generated by the remnant focusing field. This periodic perturbation produces families of parametric resonances, which form a tree of bifurcation branches. Each branch follows the tune of the unperturbed Hamiltonian. A prescription for selecting a proper unperturbed Hamiltonian will be discussed.