<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40"><head><META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=US-ASCII"><meta name=Generator content="Microsoft Word 12 (filtered medium)"><style></style></head><body bgcolor=white lang=EN-US link=blue vlink=purple style='WORD-WRAP: break-word;-webkit-nbsp-mode: space;-webkit-line-break: after-white-space' id=MailContainerBody name="Compose message area"><div class=WordSection1><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Kirk,<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'>Just ignore comments and keep sending all this. It is useful and educational.<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'> Milorad<o:p></o:p></span></p><p class=MsoNormal><span style='font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D'><o:p> </o:p></span></p><div><div style='border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0in 0in 0in'><p class=MsoNormal><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>From:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> map-l-bounces@lists.bnl.gov [mailto:map-l-bounces@lists.bnl.gov] <b>On Behalf Of </b>Kirk T McDonald<br><b>Sent:</b> Friday, March 11, 2011 11:05 AM<br><b>Cc:</b> MAP List<br><b>Subject:</b> Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></p></div></div><p class=MsoNormal><o:p> </o:p></p><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>Jim,</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>Sorry to disturb you with issues of fundamental concern for accelerator physics in general, and MAP in particular.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>So far, there have been several thought-provoking replies to my inquiries, all from people whom I would not have thought to contact.   That is, using the MAP list has evoked a useful (in my view) dialogue among people who otherwise were not in communication.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>In my view, this is what such email lists are for.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>If one doesn't like to receive such emails, one should opt out of the email list, which is, I believe, maintained by Scott Berg.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>It could be useful for the MAP collaboration to set up sublists, such as accelerator theory, targets, front end, cooling, acceleration, decay rings, MICE, design and simulation, technology development...., which would somewhat focus dialogues such as the present.   Of course, this often results in copies of emails being sent to several of the sublists, such that the total email traffic is increased, not decreased.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>But on occasion, such as the present, when one is looking for views from the broadest possible group, the master list is the right thing to use.</span><o:p></o:p></p></div><div><p class=MsoNormal> <o:p></o:p></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>--Kirk</span><o:p></o:p></p></div><div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>From:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:norem@anl.gov" title="mailto:norem@anl.gov
CTRL + Click to follow link">Jim Norem</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Sent:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> Friday, March 11, 2011 9:47 AM<o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>To:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:kirkmcd@Princeton.EDU" title="kirkmcd@Princeton.EDU">Kirk T McDonald</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Cc:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:map-l@lists.bnl.gov" title="map-l@lists.bnl.gov">MAP List</a> ; <a href="mailto:dragtnb@comcast.net" title="dragtnb@comcast.net">alex dragt</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Subject:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></p></div></div></div><div><p class=MsoNormal><o:p> </o:p></p></div><p class=MsoNormal>Do you have to send these messages to everybody? <o:p></o:p></p><div><p class=MsoNormal><o:p> </o:p></p></div><div><p class=MsoNormal>-- Jim<o:p></o:p></p></div><div><p class=MsoNormal><o:p> </o:p></p></div><div><p class=MsoNormal><o:p> </o:p></p><div><div><p class=MsoNormal>On Mar 10, 2011, at 8:29 PM, Kirk T McDonald wrote:<o:p></o:p></p></div><p class=MsoNormal><br><br><o:p></o:p></p><div name="Compose message area" id=MailContainerBody><div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>Rob,<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>1.  </span><span style='font-size:10.0pt'>"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)."</span><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'><o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>This kind of thinking is what follows from emphasis on the canonical/symplectic point of view.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>The astonishing result of Swann (now 78 years old) is that if you are willing to "think outside this box", you will find that phase volume has nothing to do with the vector potential.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>2.  It sounds like you agree with Scott Berg that it's the right thing to use (E,t) rather than (p_z,z) as "canonical" longitudinal coordinates when sampling at fixed z rather than fixed t.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>Nonetheless, it would be a service to mankind if this insight could be documented in a manner that ordinary mortals can understand.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>I note that you evade the important question of how this works in the presence of electromagnetic fields.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif"'>--Kirk<o:p></o:p></span></p></div></div><div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'><o:p> </o:p></span></p></div><div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>From:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:rdryne@lbl.gov" title="rdryne@lbl.gov">Robert D Ryne</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Sent:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> Thursday, March 10, 2011 7:33 PM<o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>To:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:alexahin@fnal.gov" title="alexahin@fnal.gov">Yuri Alexahin</a> ; <a href="mailto:kirkmcd@Princeton.EDU" title="kirkmcd@Princeton.EDU">Kirk T McDonald</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Cc:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> <a href="mailto:map-l@lists.bnl.gov" title="map-l@lists.bnl.gov">MAP List</a> ; <a href="mailto:dragtnb@comcast.net" title="dragtnb@comcast.net">alex dragt</a> ; <a href="mailto:dragtg5@comcast.net" title="dragtg5@comcast.net">Alex Dragt</a> ; <a href="mailto:dragt@physics.umd.edu" title="dragt@physics.umd.edu">Alex Dragt</a> <o:p></o:p></span></p></div><div><p class=MsoNormal style='background:whitesmoke'><b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'>Subject:</span></b><span style='font-size:10.0pt;font-family:"Tahoma","sans-serif"'> Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></p></div></div></div><div><p class=MsoNormal><o:p> </o:p></p></div><p class=MsoNormal><span class=apple-style-span><span style='font-size:13.0pt'>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></span> <o:p></o:p></p><div><p class=MsoNormal><span style='font-size:13.0pt'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>The natural quantities to be computed are called "eigen-emittances."<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>To compute them properly they need to be derived from a beam 2nd moment matrix, Sigma, formed using canonical variables.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>The eigen-emittances are invariant under linear symplectic transformations.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>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.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>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).<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'><o:p> </o:p></span></p></div><div><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><div><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><div><div><div><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><div><div><div><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><div><div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif";color:black'>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).<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif";color:black'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif";color:black'>Is there any written reference that explains this “well known” fact?<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:10.0pt;font-family:"Arial","sans-serif";color:black'> <o:p></o:p></span></p></div></div></div></blockquote></div></div></div></blockquote></div></div></div></blockquote></div></blockquote></div><div><p class=MsoNormal><span style='font-size:13.0pt'> <o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>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.<o:p></o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'><o:p> </o:p></span></p></div><div><p class=MsoNormal><span style='font-size:13.0pt'>Rob<o:p></o:p></span></p></div><div><p class=MsoNormal><o:p> </o:p></p></div><div><div><p class=MsoNormal>On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:<o:p></o:p></p></div><p class=MsoNormal><br><br><o:p></o:p></p><div><p class=MsoNormal>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><br><o:p></o:p></p><p class=MsoNormal>Folks,<o:p></o:p></p><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>There is a technical question as to how we should be calculating <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>emittance for beams in electromagnetic fields.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>The formal theory of Liouville’s theorem is clear that the invariant <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>volume in phase space is to be calculated with the canonical momentum<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>gamma m v + e A / c<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>and not the mechanical momentum m v.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>This is awkward in two ways:<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>1.   We don’t always know the vector potential of our fields<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>2.   The vector potential is subject to gauge transformations, so <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>canonical momentum is not gauge invariant.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>The second issue is disconcerting in that it suggests that phase-space <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>volume, and emittance, are not actually invariant  -- with respect to <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>gauge transformations.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Hence, it is useful to note a very old paper,<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>W.F.G. Swann, Phys. Rev. 44, 233 (1933)<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>which shows that the phase-space volume for a set of noninteracting <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>particles is the same whether or not the term e A / c is included in <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>the “momentum”.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>This result has the consequence that phase-space volume (and <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>emittance) is actually gauge invariant – although the location of a <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>volume element in space space is gauge dependent.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>---------------<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>This suggests that we could simply calculate emittances based only on <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>the mechanical momentum, and avoid having to worry about the accuracy <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>of our model for the vector potential.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Of course, our calculations are actually of rms emittance, which is a <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>better representation of the “ideal” emittance if the phase-space <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>volume is more “spherical”, and not elongated/twisted.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>It could be that the shape of the phase-space volume is better for rms <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>emittance calculation if the vector potential, in some favored gauge, <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>is included in the calculation.....<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>--Kirk<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>PS  I have placed Swann’s paper as DocDB 560<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><a href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</a><o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>user = ionization pass = mucollider1<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>See also the paper by Lemaitre that used Liouville’s theorem for <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>cosmic rays in the Earth’s atmosphere (using mechanical momentum).   <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>This may well be the earliest paper about particle beams and <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Liouville’s theorem.<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>PPS  Scott Berg notes that when one evaluates emittance at a fixed <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>plane in space, rather than at a fixed time, it is better to use the <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>“longitudinal” coordinates (E,t) rather than (P_z,z).<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Is there any written reference that explains this “well known” fact?<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>How is this prescription affected by electromagnetic fields?<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>The vector potential of even a simple rf accelerating cavity has an <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>A_z component (which is zero on axis, but nonzero off it).<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><a href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</a><o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Note that the vector potential is nonzero outside the cavity, even <o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>though the E and B fields are zero there!<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><o:p> </o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>Do we know how to include A_z in our longitudinal emittance calculations?<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>_______________________________________________<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal>MAP-l mailing list<o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><o:p></o:p></p></blockquote><blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'><p class=MsoNormal><a href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</a><o:p></o:p></p></blockquote><p class=MsoNormal>_______________________________________________<br>MAP-l mailing list<br><a href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</a><br><a href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</a><o:p></o:p></p></div></div><p class=MsoNormal><o:p> </o:p></p></div><p class=MsoNormal>_______________________________________________<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<o:p></o:p></p></div><p class=MsoNormal><o:p> </o:p></p></div></div></body></html>