arXiv2026
Charged particle beams in equilibrium with their collective fields are represented by self-consistent phase space distributions. As the starting point, we assume the external focusing forces to act continuously on the beam. In the collisionless mean-field approximation, a broad class of stationary self-consistent particle phase space distributions can then be constructed. The method is reviewed how to determine the single-particle Hamiltonian of the focusing system for a given phase space density function. Subsequently, canonical transformations are used to relate this description to a beam passing through a non-continuous transport system. The correspondence is exact for the linear K-V model and provides an approximate matching prescription for distributions with nonlinear space-charge forces. The transverse Hamiltonian generally changes with the focusing strength, including in a rotationally symmetric periodic solenoid channel. Our computer simulations, presented in the second part of the paper, indicate very small rms-emittance changes for appropriately matched beams in the continuous and periodic solenoid channels investigated. The simulation for a periodic quadrupole channel yields a small persistent increase of the rms-emittance, whose physical interpretation requires control of numerical errors. Revised version of: Particle~Accelerators 39, 219--249 (1992)}