Analytic Nonlinear Methods for Beam Optics

The range of meaning for the word “analytical” extends from a hand-derived formula to be used by hand, to a computer-derived formula to be used by hand (eg. with a symbolic manipulation program), to a computer-derived expression to be seen only by a computer. There is an increasing degree of complexity as one proceeds through these types. The hand-derived/hand-used is especially important for the first stages of beamline design, the computer-derived/hand-used is especially important for use in the operation and tuning of an existing beamline, and computer-derived/computer-used is useful in the intermediate and final stages of beamline design. We will describe five examples, covering a full range of complexity. From least to most complex these examples will be: • use of similarity transformations (FFTB design, SLC diagnosis) • statistical maps (SSC smear and tune-shift) • aberrations (SLC upgrade) • resonance basis and nPB tracking (PEP-II design) • kick factorization (possibly LHC).