Unified treatment of collective instabilities and nonlinear beam dynamics

Nonlinear dynamics deals with parametric resonances and diffusion, which are usually beam-intensity independent and rely on a particle Hamiltonian. Collective instabilities deal with beam coherent motion, where the Vlasov equation is frequently used in conjunction with a beam-intensity dependent Hamiltonian. We address the questions: Are the two descriptions the same? Are collective instabilities the results of encountering parametric resonances whose driving force is intensity dependent? The space-charge dominated beam governed by the Kapchinskij-Vladimirskij (KV) envelope equation is used as an example.