Normal form via tracking or beam data

Normal Form is a powerful tool to analyse the nonlinear content of a complicated system like an accelerator with its many thousands of high order multipolar errors. This technique needs as input a mapping from the initial to the final coordinates. Unfortunately, a map of an accelerator is not known beforehand but has to be determined for a complete lattice of the accelerator including all calculated and/or measured error tables of the guiding and focussing magnet elements. It is however possible to obtain turn-by-turn position data of a kicked beam for many turns. In various experiments at existing accelerators it has been shown that these data are equivalent to tracking data produced by simulation programs. In this report we will demonstrate how tracking data can be used to determine, with excellent precision, the coefficients of the generating function. The well tested Normal Form tools can then be used to construct the underlying Hamiltonian and the map. The essential tool is the recently developed frequency analysis which allows for a very precise determination of the tunes using beam or tracking data.