Hamiltonian formalism for space charge dominated beams in a uniform focusing channel

We employ the Kapchinskij-Vladimirskij envelope Hamiltonian [I. M. Kapchinskij and V. V. Vladimirskij, in 2 The Proceedings of the 9th International Conference on High Energy Accelerators, edited by L. Kowarski (CERN, Geneva, 1959), p. 274] to describe the envelope evolution and the particle Hamiltonian to describe particle motion in a space charge dominated beam. Properties of the envelope function in a mismatched uniform focusing channel are studied. Parametric resonances of the particle Hamiltonian due to envelope oscillations of a mismatched beam are studied. We find that the Hamiltonian dynamics depends only on a single effective space charge parameter, the ratio of the space charge perveance to the focusing strength. The onset of global chaos exhibits a first order phase-transition-like behavior when the amplitude of envelope oscillations for a mismatched beam is larger than a critical value. This global chaos can greatly enhance the halo formation. The relation between the critical envelope mismatch for the halo formation and the effective space charge parameter is numerically obtained. Possible experiments are suggested