<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Dear Kirk,<div><br></div><div>Gauge transformations are symplectic maps. See Exercises 6.2.8 and 6.5.3 in the book "Lie Methods ...", which can be downloaded from the Web site</div><div><br></div><div><a href="http://www.physics.umd.edu/dsat/">http://www.physics.umd.edu/dsat/</a></div><div><br></div><div>By construction, eigen emittances are invariant under linear symplectic transformations, and therefore invariant under gauge transformations in the linear approximation. Note the words "eigen emittances"! Eigen emittances are not the same as rms emittances. See the papers</div><div><br></div><div><div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">A. Dragt, F. Neri, et al., "LIE ALGEBRAIC TREATMENT OF LINEAR AND NONLINEAR BEAM DYNAMICS",</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">Annual Review of Nuclear and Particle Science 38, p. 455 (1988).</div></div><div><font class="Apple-style-span" size="5"><span class="Apple-style-span" style="font-size: 18px; "><br></span></font></div><div><font class="Apple-style-span" size="5"><span class="Apple-style-span" style="font-size: 18px; "><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">A. Dragt, R. Gluckstern, et al., "THEORY OF EMITTANCE INVARIANTS", Lecture Notes in</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">Physics 343: Proceedings of the Joint US-CERN Capri School on Accelerator Physics,</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">Springer Verlag (1989).</div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; "><br></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; "><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">A. Dragt, F. Neri, et al., "GENERAL MOMENT INVARIANTS FOR LINEAR HAMILTONIAN SYSTEMS", </div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; ">Physical Review A, 45, p. 2572 (1992).</div></div></span></font></div></div><div><br></div><div><br></div><div>Rather, eigen emittances tell you what rms emittances can be achieved by proper beam manipulation (using symplectic elements in the linear approximation). See Chapter 26 of Lie Methods ....</div><div><br></div><div>Every symplectic map can be factorized into linear and nonlinear parts, all of which are symplectic. The linear part of a symplectic map is described by a symplectic matrix. See Sections 7.6 through 7.8 of Lie Methods .... [Every map (in an even number of variables) can be factorized into symplectic and nonsymplectic parts. See Chapter 22 of Lie methods.] As related in Section 1.1.2 of Lie Methods..., accelerators were originally designed using only linear linear symplectic maps (symplectic matrices). In fact, it was often not recognized that even the so called linear beam-line elements such as drifts, quads, and bends also have nonlinear parts. Karl Brown was the first to include quadratic effects, which made it possible to treat sextupoles in the lowest nonlinear approximation. With the advent of Lie methods and Truncated Power Series Algebra (TPSA) it is now possible to work to quite high order, at least for idealized elements excluding s dependences and fringe fields. We are currently working (see below) to extend these methods to realistic beam-line elements including fringe fields and high-order multipole effects.</div><div><br></div><div>Although an approximation, it is always good to begin with a linear (symplectic matrix) design. But then one must recognize that nonlinear corrections can be important. For example, for many years, SLAC failed to included the third-order nonlinear terms in drifts when trying to simulate the B-factory dynamic aperture. To their amazement, they found that the agreement between simulation and experiment improved when they finally did so. In the cases of solenoids and quadrupoles it is important to recognize that third-order fringe-field effects can be important. Indeed, in the case of electron microscopes, solenoid third-order fringe-field effects are the chief source of aberrations. And, in the case of microprobes and beam telescopes, quadrupole third-order fringe-field effects are the chief source of aberrations. Due to their complicated nature, not much is known about the effects of fringe fields for realistic dipoles. This will change with the fruition of the work on curved elements, also sketched below.</div><div><br></div><div>With regard to the effect of vector potential terms, it is always possible to select a gauge such that the vector potential vanishes in the field-free regions. Indeed, when B=0, we must have curl A=0, and therefore A is a gradient of a scalar field, and this scalar field can be used to gauge transform A to zero, etc. We have developed surface methods which take E and B field data as input and produce as output a vector potential. This vector potential has the property that it decays to zero outside the element. See the paper</div><div><br></div><div><span class="Apple-style-span" style="font-family: arial, helvetica, sans-serif; color: rgb(50, 50, 50); -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px; "><h2 style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0.4em; padding-right: 0px; padding-bottom: 0.5em; padding-left: 15px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; font-family: arial, helvetica, sans-serif; font-size: 1em; font-weight: bold; color: rgb(1, 93, 170); line-height: 1.3em; ">Phys. Rev. ST Accel. Beams 13, 064001 (2010) [17 pages]</h2><h1 style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; padding-top: 0px; padding-right: 0px; padding-bottom: 0.4em; padding-left: 15px; border-top-width: 0px; border-right-width: 0px; border-bottom-width: 0px; border-left-width: 0px; border-style: initial; border-color: initial; font-family: arial, helvetica, sans-serif; font-size: 1.5em; font-weight: bold; color: rgb(1, 93, 170); line-height: 1.3em; ">Accurate transfer maps for realistic beam-line elements: Straight elements</h1></span></div><div><br></div><div>which deals with straight magnetic elements. This paper also cites the work of Dan Abell, which does analogous things for realistic RF cavities. For the beginning of work on curved elements, see the attachment</div><div><br></div><div><br></div><div></div></body></html>