Transport matrices for nonlinear lattice functions

We review some of the methods currently used to reliably calculate quasi-invariants and we make transparent the relation between our previous work and part of this article, with the Lie algebraic technique. We construct the explicit relation between them for the problem of transverse motion of particles in a storage ring. Here, the onset of distortions of phase space due to amplitude and chromatic effects is well described by periodic functions in addition to the Courant-Snyder functions α, β and γ. To use the periodic functions at the design stage of the machine, it is necessary to compute these functions efficiently. For this purpose we also present a convenient list of transport matrices for the periodic functions.