Symplectic Mapping in Systems with Three-Dimensional Magnetic Fields

Peace Chang ； S. Y. Lee ； Y. T. Yan

《Chinese Journal of Physics》 44卷2期 (2006/04) Pp. 150-164

並列摘要

A differential algebraic integration algorithm is developed for symplectic particle mapping through three-dimensional (3D) magnetic field configurations. The algorithm employs a canonical transformation to eliminate the linear part of the Hamiltonian. This is equivalent to solving the equations of motion with a Lorentz force for a reference orbit. The new Hamiltonian is then Taylor expanded around the reference orbit. Making thin slices along the longitudinal coordinate and using differential algebras, one can calculate and store the energy-dependent reference orbit, the coordinate-dependent vector potential, and thus the Hamiltonian as well as the symplectic map in a Lie transformation form at each slice. The section map from the entrance to the exit in the 3D magnetic field structure is then concatenated through the slices. This procedure can be used to obtain particle transfer maps for insertion devices, solenoids, fringe fields of dipoles and quadruples, and other complicated magnetic configurations.