ABSTRACT
In this chapter, the authors aims to study the action of transfer matrices on particles by looking in detail to what happens to entire regions of phase space as they are transported. This is important because the beam in an accelerator is just such a region, and of course the authors want to make sure that at any time, this region is within the beam pipe. In order to study the motion of ensembles of particles under linear transformations, it is useful to characterize them by certain simple geometric forms in which the particles are contained and requiring only few parameters. The two most useful such forms are the polygon and the ellipse. With the help of the Differential Algebraic technique, it is straightforward to include parameter dependence of the Twiss parameters, which can introduce beating due to momentum deviation or quadrupole errors.
