<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="font-size: 17px; ">I have not yet read the papers mentioned. But here are some brief comments. Alex Dragt and I (cc to Alex) have been thinking about this a lot in the past months.</span><div style="font-size: 17px; "><br></div><div style="font-size: 17px; ">The natural quantities to be computed are called "eigen-emittances."</div><div style="font-size: 17px; ">To compute them properly they need to be derived from a beam 2nd moment matrix, Sigma, formed using canonical variables.</div><div style="font-size: 17px; ">The eigen-emittances are invariant under linear symplectic transformations.</div><div style="font-size: 17px; "><br></div><div style="font-size: 17px; ">The eigen-emittances can be computed in various ways, but the simplest is to compute the eigen-values of J Sigma, where J is the fundamental symplectic 2-form; the eigen-emittances are the modulii of the eigen-values of J Sigma (which are pure imaginary and in +/- pairs). If one is interested in calculating the symplectic matrix that transforms Sigma to Williamson normal form, Alex Dragt has an algorithm to do this and has implemented it in the MaryLie code.</div><div style="font-size: 17px; "><br></div><div style="font-size: 17px; ">Though the entries of Sigma will depend on the choice of gauge, the eigen-emittances themselves are gauge invariant. We can't just set the vector potential to zero inside elements where it is nonzero, and expect to calculate the correct eigen-emittances (as was suggested below).</div><div style="font-size: 17px; "><br></div><div style="font-size: 17px; "><blockquote type="cite" style="font-size: medium; "><div text="#000000" bgcolor="#ffffff"><blockquote cite="mid:98FEE0D2849743468CEB0449A30CF803@mumu30" type="cite"><div dir="ltr"><div style="font-family: Arial; color: rgb(0, 0, 0); font-size: 10pt; "><div style="font-style: normal; display: inline; font-family: Calibri; color: rgb(0, 0, 0); font-size: small; font-weight: normal; text-decoration: none; "><blockquote cite="mid:608291C1C4744041A10D5279278A9353@mumu30" type="cite"><div dir="ltr"><div style="font-family: Arial; color: rgb(0, 0, 0); font-size: 10pt; "><div style="font-style: normal; display: inline; font-family: Calibri; color: rgb(0, 0, 0); font-size: small; font-weight: normal; text-decoration: none; "><blockquote cite="mid:468B48A3C96B4BA3AA66387F9E650168@mumu30" type="cite"><div dir="ltr"><div style="font-family: Arial; color: rgb(0, 0, 0); font-size: 10pt; "><div>PPS  Scott Berg notes that when one evaluates emittance at a fixed plane in space, rather than at a fixed time, it is better to use the “longitudinal” coordinates (E,t) rather than (P_z,z).</div><div> </div><div>Is there any written reference that explains this “well known” fact?</div><div> </div></div></div></blockquote></div></div></div></blockquote></div></div></div></blockquote></div></blockquote></div><div style="font-size: 17px; "> </div><div style="font-size: 17px; ">The above follows directly from whether we use the time t as the independent variable or the Cartesian coordinate z as the independent variable. When using the time, the longitudinal variables are (z,p_{z,canonical}). When using z, the longitudinal variables are (t, -E) where t is arrival time at location z, and where E is the total energy of a particle when it reaches location z, i.e. E=\gamma m c^2 + q \Phi.</div><div style="font-size: 17px; "><br></div><div style="font-size: 17px; ">Rob</div><div><br></div><div><div>On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Hi Kirk,<br><br>Thank you for digging out these interesting papers.<br>Of course the Poincare invariants remain the same no matter what momenta are used.<br>But this is not what we calculate from tracking or measurement data using standard definition.<br>So a clarification is still needed of what and how we should calculate.<br><br>Yuri<br><br>----- Original Message -----<br>From: Kirk T McDonald <<a href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</a>><br>Date: Thursday, March 10, 2011 4:09 pm<br>Subject: [MAP] Liouville's theorem and electromagnetic fields<br>To: MAP List <<a href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</a>><br>Cc: Kirk McDonald <<a href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</a>><br><br><br><blockquote type="cite">Folks,<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">There is a technical question as to how we should be calculating <br></blockquote><blockquote type="cite">emittance for beams in electromagnetic fields.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The formal theory of Liouville’s theorem is clear that the invariant <br></blockquote><blockquote type="cite">volume in phase space is to be calculated with the canonical momentum<br></blockquote><blockquote type="cite">gamma m v + e A / c<br></blockquote><blockquote type="cite">and not the mechanical momentum m v.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">This is awkward in two ways:<br></blockquote><blockquote type="cite">1.   We don’t always know the vector potential of our fields<br></blockquote><blockquote type="cite">2.   The vector potential is subject to gauge transformations, so <br></blockquote><blockquote type="cite">canonical momentum is not gauge invariant.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The second issue is disconcerting in that it suggests that phase-space <br></blockquote><blockquote type="cite">volume, and emittance, are not actually invariant  -- with respect to <br></blockquote><blockquote type="cite">gauge transformations.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Hence, it is useful to note a very old paper,<br></blockquote><blockquote type="cite">W.F.G. Swann, Phys. Rev. 44, 233 (1933)<br></blockquote><blockquote type="cite">which shows that the phase-space volume for a set of noninteracting <br></blockquote><blockquote type="cite">particles is the same whether or not the term e A / c is included in <br></blockquote><blockquote type="cite">the “momentum”.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">This result has the consequence that phase-space volume (and <br></blockquote><blockquote type="cite">emittance) is actually gauge invariant – although the location of a <br></blockquote><blockquote type="cite">volume element in space space is gauge dependent.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">---------------<br></blockquote><blockquote type="cite">This suggests that we could simply calculate emittances based only on <br></blockquote><blockquote type="cite">the mechanical momentum, and avoid having to worry about the accuracy <br></blockquote><blockquote type="cite">of our model for the vector potential.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Of course, our calculations are actually of rms emittance, which is a <br></blockquote><blockquote type="cite">better representation of the “ideal” emittance if the phase-space <br></blockquote><blockquote type="cite">volume is more “spherical”, and not elongated/twisted.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">It could be that the shape of the phase-space volume is better for rms <br></blockquote><blockquote type="cite">emittance calculation if the vector potential, in some favored gauge, <br></blockquote><blockquote type="cite">is included in the calculation.....<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">--Kirk<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">PS  I have placed Swann’s paper as DocDB 560<br></blockquote><blockquote type="cite"><a href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</a><br></blockquote><blockquote type="cite">user = ionization pass = mucollider1<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">See also the paper by Lemaitre that used Liouville’s theorem for <br></blockquote><blockquote type="cite">cosmic rays in the Earth’s atmosphere (using mechanical momentum).   <br></blockquote><blockquote type="cite">This may well be the earliest paper about particle beams and <br></blockquote><blockquote type="cite">Liouville’s theorem.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">PPS  Scott Berg notes that when one evaluates emittance at a fixed <br></blockquote><blockquote type="cite">plane in space, rather than at a fixed time, it is better to use the <br></blockquote><blockquote type="cite">“longitudinal” coordinates (E,t) rather than (P_z,z).<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Is there any written reference that explains this “well known” fact?<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">How is this prescription affected by electromagnetic fields?<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The vector potential of even a simple rf accelerating cavity has an <br></blockquote><blockquote type="cite">A_z component (which is zero on axis, but nonzero off it).<br></blockquote><blockquote type="cite"><a href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</a><br></blockquote><blockquote type="cite">Note that the vector potential is nonzero outside the cavity, even <br></blockquote><blockquote type="cite">though the E and B fields are zero there!<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Do we know how to include A_z in our longitudinal emittance calculations?<br></blockquote><blockquote type="cite">_______________________________________________<br></blockquote><blockquote type="cite">MAP-l mailing list<br></blockquote><blockquote type="cite"><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br></blockquote><blockquote type="cite"><a href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</a><br></blockquote>_______________________________________________<br>MAP-l mailing list<br><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br>https://lists.bnl.gov/mailman/listinfo/map-l<br></div></blockquote></div><br></body></html>