Linear and nonlinear decoupling using time-dependent transformations

Linear coupling in a storage ring is conveniently analyzed in terms of transformations that put the single-turn map into block-diagonal form. Such a transformation allows us to define new variables, in which the dynamics are uncoupled. In this paper it is shown how a similar approach may be taken to nonlinear coupling, but that to decouple the map completely one needs to use a time-dependent canonical transformation. In Sec. III, we present a numerical example, based upon the analysis presented in previous sections, of a nonlinear transformation. In part for pedagogical reasons, and in part to make our use of notation clear, in Appendix A we reproduce the theory, along with a numerical example, of the well-known result for a linear transformation.