Transverse motion of single particles in accelerators

Any accelerator must have focusing elements, usually magnetic quadrupoles, which hold individual particles close to an ideal central orbit. The purpose of this chapter is to analyse the transverse motion of particles and beams in such systems, with particular emphasis on circular machines. A particle not precisely on the central orbit is subject to a restoring force tending to reduce the deviation and, as a result, the particle oscillates around the central orbit. These oscillations are usually called betatron oscillations when they occur in a circular machine, and we will use the same terminology even for motion in any beam transport system