<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><font class="Apple-style-span" size="5"><span class="Apple-style-span" style="font-size: 18px;">For the use of t and p_t as canonical variables, see Sections 1.6 and 1.7 and the associated Exercises in my Lie Methods... book. The Web site is</span></font><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;"><a href="http://www.physics.umd.edu/dsat/">http://www.physics.umd.edu/dsat/</a></span></font></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;">Also, given the fields E and B on a grid, we are beginning to have codes that produce the associated vector potential A using surface methods.</span></font></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;">Best,</span></font></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;">Alex<br></span></font><div><br></div><div><br><div><div>On Mar 10, 2011, at 4:48 PM, Kirk T McDonald wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Folks,<br><br>I have added Alex' paper to DocDB 560. See Appendix A.<br><br>It is gratifying to see that the fact that Liouville's theorem holds for both mechanical and canonical phase space is "well known to those who know".<br><br>The challenge now is to learn how best to use the "freedom" offered to us by this apparently nonintuitive result.<br><br>--Kirk<br><br>--------------------------------------------------<br>From: "alex dragt" <<a href="mailto:dragtnb@comcast.net">dragtnb@comcast.net</a>><br>Sent: Thursday, March 10, 2011 9:23 PM<br>To: "Don Summers" <<a href="mailto:summers@phy.olemiss.edu">summers@phy.olemiss.edu</a>><br>Cc: "Robert D Ryne" <<a href="mailto:rdryne@lbl.gov">rdryne@lbl.gov</a>>; "Yuri Alexahin" <<a href="mailto:alexahin@fnal.gov">alexahin@fnal.gov</a>>; "Kirk T McDonald" <<a href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</a>>; "Alex Dragt" <<a href="mailto:dragtg5@comcast.net">dragtg5@comcast.net</a>>; "MAP List" <<a href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</a>>; "Alex Dragt" <<a href="mailto:dragt@physics.umd.edu">dragt@physics.umd.edu</a>><br>Subject: Re: [MAP] Liouville's theorem and electromagnetic fields<br><br><blockquote type="cite">Dear all,<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">The fact that Liouville's theorem holds in both mechanical and canonical phase space is also proved in<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">A. Dragt<br></blockquote><blockquote type="cite">SOLAR CYCLE MODULATION OF THE RADIATION BELT PROTON FLUX, J.<br></blockquote><blockquote type="cite">Geophys. Res. 76: 2312-2344 (1971)<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">also done in the context of the Van Allen Radiation, and hence for motion in the Earth's Magnetic Field.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">But we are interested in more than Liouville's theorem. Also note that gauge transformations are symplectic maps, and hence do not affect the eigen emitances. See the book Lie Methods ... available at the Web site<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><a href="http://www.physics.umd.edu/dsat/">http://www.physics.umd.edu/dsat/</a><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Best,<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Alex<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">On Mar 10, 2011, at 3:55 PM, Don Summers wrote:<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite">The exact reference for Swann's paper is<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">W. F. G. Swann, Application of Liouville's Theorem to Electron Orbits in the Earth's Magnetic Field,<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Phys. Rev. 44, 224–227 (1933)<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><a href="http://prola.aps.org/abstract/PR/v44/i3/p224_1">http://prola.aps.org/abstract/PR/v44/i3/p224_1</a><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Best,<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Don<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">On Thu, 10 Mar 2011 16:33:10 -0800, Robert D Ryne wrote<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">I have not yet read the papers mentioned. But here are some brief<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">comments. Alex Dragt and I (cc to Alex) have been thinking about<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">this a lot in the past months.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The natural quantities to be computed are called "eigen-emittances."<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">To compute them properly they need to be derived from a beam 2nd<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">moment matrix, Sigma, formed using canonical variables.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The eigen-emittances are invariant under linear symplectic<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">transformations.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The eigen-emittances can be computed in various ways, but the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">simplest is to compute the eigen-values of J Sigma, where J is the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">fundamental symplectic 2-form; the eigen-emittances are the modulii<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">of the eigen- values of J Sigma (which are pure imaginary and in +/-<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">pairs). If one is interested in calculating the symplectic matrix<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">that transforms Sigma to Williamson normal form, Alex Dragt has an<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">algorithm to do this and has implemented it in the MaryLie code.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Though the entries of Sigma will depend on the choice of gauge, the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">eigen-emittances themselves are gauge invariant. We can't just set<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the vector potential to zero inside elements where it is nonzero,<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">and expect to calculate the correct eigen-emittances (as was<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">suggested below).<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">PPS Scott Berg notes that when one evaluates emittance at a<br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">fixed plane in space, rather than at a fixed time, it is better<br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">to use the [WINDOWS-1252?]“longitudinal” coordinates (E,t) rather than (P_z,z).<br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Is there any written reference that explains this [WINDOWS-1252?]“well [WINDOWS-<br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">1252?]known”<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">fact?<br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The above follows directly from whether we use the time t as the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">independent variable or the Cartesian coordinate z as the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">independent variable. When using the time, the longitudinal<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">variables are<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">(z,p_{z,canonical}). When using z, the longitudinal variables are (t,<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">- E) where t is arrival time at location z, and where E is the<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">total energy of a particle when it reaches location z, i.e.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">E=\gamma m c^2 + q \Phi.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Rob<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Hi Kirk,<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Thank you for digging out these interesting papers.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Of course the Poincare invariants remain the same no matter what<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">momenta are used.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">But this is not what we calculate from tracking or measurement data<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">using standard definition.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">So a clarification is still needed of what and how we should<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">calculate.<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Yuri<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">----- Original Message -----<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">From: Kirk T McDonald <<a href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</a>><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Date: Thursday, March 10, 2011 4:09 pm<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Subject: [MAP] Liouville's theorem and electromagnetic fields<br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">To: MAP List <<a href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</a>><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Cc: Kirk McDonald <<a href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</a>><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Folks,<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">There is a technical question as to how we should be calculating<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">emittance for beams in electromagnetic fields.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The formal theory of [WINDOWS-1252?]Liouville’s theorem is clear that the invariant<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">volume in phase space is to be calculated with the canonical momentum<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">gamma m v + e A / c<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">and not the mechanical momentum m v.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">This is awkward in two ways:<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">1. We [WINDOWS-1252?]don’t always know the vector potential of our fields<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">2. The vector potential is subject to gauge transformations, so<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">canonical momentum is not gauge invariant.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The second issue is disconcerting in that it suggests that phase-<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">space<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">volume, and emittance, are not actually invariant -- with respect to<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">gauge transformations.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Hence, it is useful to note a very old paper,<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">W.F.G. Swann, Phys. Rev. 44, 233 (1933)<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">which shows that the phase-space volume for a set of noninteracting<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">particles is the same whether or not the term e A / c is included in<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the [WINDOWS-1252?]“momentum”.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">This result has the consequence that phase-space volume (and<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">emittance) is actually gauge invariant [WINDOWS-1252?]– although the location of a<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">volume element in space space is gauge dependent.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">---------------<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">This suggests that we could simply calculate emittances based only on<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the mechanical momentum, and avoid having to worry about the accuracy<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">of our model for the vector potential.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Of course, our calculations are actually of rms emittance, which is a<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">better representation of the [WINDOWS-1252?]“ideal” emittance if the phase-space<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">volume is more [WINDOWS-1252?]“spherical”, and not elongated/ twisted.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">It could be that the shape of the phase-space volume is better for<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">rms<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">emittance calculation if the vector potential, in some favored gauge,<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">is included in the calculation.....<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">--Kirk<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">PS I have placed [WINDOWS-1252?]Swann’s paper as DocDB 560<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><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></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">user = ionization pass = mucollider1<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">See also the paper by Lemaitre that used [WINDOWS-1252?]Liouville’s theorem for<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">cosmic rays in the [WINDOWS-1252?]Earth’s atmosphere (using mechanical momentum).<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">This may well be the earliest paper about particle beams and<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">[WINDOWS-1252?]Liouville’s theorem.<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">PPS Scott Berg notes that when one evaluates emittance at a fixed<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">plane in space, rather than at a fixed time, it is better to use the<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">[WINDOWS-1252?]“longitudinal” coordinates (E,t) rather than (P_z,z).<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Is there any written reference that explains this [WINDOWS-1252?]“well [WINDOWS-<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">1252?]known” fact?<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">How is this prescription affected by electromagnetic fields?<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">The vector potential of even a simple rf accelerating cavity has an<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">A_z component (which is zero on axis, but nonzero off it).<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><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></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Note that the vector potential is nonzero outside the cavity, even<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">though the E and B fields are zero there!<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Do we know how to include A_z in our longitudinal emittance<br></blockquote></blockquote></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><blockquote 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