Equations of Motion in Accelerators

Abstract

A typical accelerator uses a sequence of various types of magnets separated by sections of free space (so-called drifts) to control the motion of the particle beam. To specify the Hamiltonian (5.23) we need to know the vector potential, $A_s$, for these magnets. We evaluate the fields and Hamiltonian for the major magnet types assuming that the field profiles are uniform over their length. Often in analysis and simulations, one has to take into account that at the end points of the magnets different field geometries appear, called fringe fields. The impact of these fields are usually treated as highly localized corrections which are calculated separately from the bulk of the magnet, and involve higher order terms that we will simply neglect in this chapter. When fringe fields are weak they can be treated as field errors, which are covered in Chap. 8.