Normalization of the parameterized Courant-Snyder matrix for symplectic factorization of a parameterized Taylor map

The transverse motion of charged particles in a circular accelerator can be well represented by a one-turn high-order Taylor map. For particles without energy deviation, the one-turn Taylor map is a four-dimensional polynomial of four variables. The four variables are the transverse canonical coordinates and their conjugate moments. To include the energy deviation (off-momentum) effects, the map has to be parameterized with a smallness factor representing the off-momentum, and so the Taylor map becomes a four-dimensional polynomial of five variables. It is for this type of parameterized Taylor map that a method is presented for converting it into a parameterized Dragt-Finn factorization map.<>