ABSTRACT
This chapter describes, a modern method, the Differential Algebraic (DA) technique, for computing the transfer map to arbitrary order. The Taylor coefficients belonging to terms of orders 2 or higher are usually called aberrations, as they describe corrections to the linear part of the map that are usually small if the phase space variables are small. In most cases, the freedom of the aberration coefficients is severely restricted by the presence of a variety of symmetries. An application of the DA concatenator is the transformation of coordinates among different codes for the study of the dynamics of beams. While many codes work in the discussed curvilinear canonical coordinates, other codes use the slope instead of the normalized momentum. In certain cases, Cartesian coordinates are used, which may be a better choice for certain elements, for example, wigglers, but are usually not very well suited for a discussion of the properties of beamlines.
