One dimensional motion in a thick sextupole-a comparison of tracking methods with the exact solution

It is shown that the comparison of Taylor-series mappings with the exact solution of the one-dimensional equations of motion for a charged particle in a sextupole permits the derivation of explicit expressions for the convergence. The existence of a limiting step size is found. It depends in general on the initial conditions (x₀, u₀) but not on the order of the mapping. These results are similar for all multipolar elements, and they restrict tracking of accelerator lattices by Taylor mappings to a maximum step size. Hence, concatenation of too many nonlinear elements will become divergent even for arbitrary high orders in the expansion of a map. This restriction of concatenation applies also to kick codes, although the zero length of a single element guarantees that the mapping does not diverge element by element. In practice, the convergence of a concatenated map can be tested by comparing results for different step sizes