<html><head><base href="x-msg://98/"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">I have a suggestion. Why don't we get another list set up for people who want to hash out the subtleties of emittance. We all know it's important, but many of us don't have time to dig into the details just now.<div><br></div><div><span class="Apple-tab-span" style="white-space:pre"> </span>Dan<br><div><br><div><div>On Mar 11, 2011, at 11:09 AM, Milorad Popovic wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><span class="Apple-style-span" style="border-collapse: separate; font-family: Helvetica; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; font-size: medium; "><div bgcolor="white" lang="EN-US" link="blue" vlink="purple" id="MailContainerBody" name="Compose message area" style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div class="WordSection1" style="page: WordSection1; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125); ">Kirk,<o:p></o:p></span></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125); ">Just ignore comments and keep sending all this. It is useful and educational.<o:p></o:p></span></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125); ">Milorad<o:p></o:p></span></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 11pt; font-family: Calibri, sans-serif; color: rgb(31, 73, 125); "><o:p> </o:p></span></div><div><div style="border-right-style: none; border-bottom-style: none; border-left-style: none; border-width: initial; border-color: initial; border-top-style: solid; border-top-color: rgb(181, 196, 223); border-top-width: 1pt; padding-top: 3pt; padding-right: 0in; padding-bottom: 0in; padding-left: 0in; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">From:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:map-l-bounces@lists.bnl.gov" style="color: blue; text-decoration: underline; ">map-l-bounces@lists.bnl.gov</a><span class="Apple-converted-space"> </span>[mailto:map-l-bounces@lists.bnl.gov]<span class="Apple-converted-space"> </span><b>On Behalf Of<span class="Apple-converted-space"> </span></b>Kirk T McDonald<br><b>Sent:</b><span class="Apple-converted-space"> </span>Friday, March 11, 2011 11:05 AM<br><b>Cc:</b><span class="Apple-converted-space"> </span>MAP List<br><b>Subject:</b><span class="Apple-converted-space"> </span>Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></div></div></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">Jim,</span><o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">In my view, this is what such email lists are for.</span><o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "> <o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">--Kirk</span><o:p></o:p></div></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><o:p> </o:p></span></div></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">From:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:norem@anl.gov" title="mailto:norem@anl.gov
CTRL + Click to follow link" style="color: blue; text-decoration: underline; ">Jim Norem</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Sent:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span>Friday, March 11, 2011 9:47 AM<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">To:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:kirkmcd@Princeton.EDU" title="kirkmcd@Princeton.EDU" style="color: blue; text-decoration: underline; ">Kirk T McDonald</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Cc:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:map-l@lists.bnl.gov" title="map-l@lists.bnl.gov" style="color: blue; text-decoration: underline; ">MAP List</a><span class="Apple-converted-space"> </span>;<span class="Apple-converted-space"> </span><a href="mailto:dragtnb@comcast.net" title="dragtnb@comcast.net" style="color: blue; text-decoration: underline; ">alex dragt</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Subject:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span>Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></div></div></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Do you have to send these messages to everybody?<o:p></o:p></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">-- Jim<o:p></o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">On Mar 10, 2011, at 8:29 PM, Kirk T McDonald wrote:<o:p></o:p></div></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><br><br><o:p></o:p></div><div name="Compose message area" id="MailContainerBody"><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">Rob,<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">1. <span class="Apple-converted-space"> </span></span><span style="font-size: 10pt; ">"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: 10pt; font-family: Arial, sans-serif; "><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; ">--Kirk<o:p></o:p></span></div></div></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><o:p> </o:p></span></div></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">From:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:rdryne@lbl.gov" title="rdryne@lbl.gov" style="color: blue; text-decoration: underline; ">Robert D Ryne</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Sent:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span>Thursday, March 10, 2011 7:33 PM<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">To:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:alexahin@fnal.gov" title="alexahin@fnal.gov" style="color: blue; text-decoration: underline; ">Yuri Alexahin</a><span class="Apple-converted-space"> </span>;<span class="Apple-converted-space"> </span><a href="mailto:kirkmcd@Princeton.EDU" title="kirkmcd@Princeton.EDU" style="color: blue; text-decoration: underline; ">Kirk T McDonald</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Cc:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span><a href="mailto:map-l@lists.bnl.gov" title="map-l@lists.bnl.gov" style="color: blue; text-decoration: underline; ">MAP List</a><span class="Apple-converted-space"> </span>;<span class="Apple-converted-space"> </span><a href="mailto:dragtnb@comcast.net" title="dragtnb@comcast.net" style="color: blue; text-decoration: underline; ">alex dragt</a><span class="Apple-converted-space"> </span>;<span class="Apple-converted-space"> </span><a href="mailto:dragtg5@comcast.net" title="dragtg5@comcast.net" style="color: blue; text-decoration: underline; ">Alex Dragt</a><span class="Apple-converted-space"> </span>;<span class="Apple-converted-space"> </span><a href="mailto:dragt@physics.umd.edu" title="dragt@physics.umd.edu" style="color: blue; text-decoration: underline; ">Alex Dragt</a><o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; background-image: initial; background-attachment: initial; background-origin: initial; background-clip: initial; background-color: rgb(245, 245, 245); "><b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; ">Subject:</span></b><span style="font-size: 10pt; font-family: Tahoma, sans-serif; "><span class="Apple-converted-space"> </span>Re: [MAP] Liouville's theorem and electromagnetic fields<o:p></o:p></span></div></div></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span class="apple-style-span"><span style="font-size: 13pt; ">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></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "><o:p> </o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">The natural quantities to be computed are called "eigen-emittances."<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">The eigen-emittances are invariant under linear symplectic transformations.<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "><o:p> </o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "><o:p> </o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "><o:p> </o:p></span></div></div><div><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div><div><div><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div><div><div><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; 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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; color: black; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; color: black; ">Is there any written reference that explains this “well known” fact?<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 10pt; font-family: Arial, sans-serif; color: black; "> <o:p></o:p></span></div></div></div></div></blockquote></div></div></div></blockquote></div></div></div></blockquote></div></blockquote></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "> <o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">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></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; "><o:p> </o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><span style="font-size: 13pt; ">Rob<o:p></o:p></span></div></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></div><div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:<o:p></o:p></div></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><br><br><o:p></o:p></div><div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">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" style="color: blue; text-decoration: underline; ">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" style="color: blue; text-decoration: underline; ">map-l@lists.bnl.gov</a>><br>Cc: Kirk McDonald <<a href="mailto:kirkmcd@Princeton.EDU" style="color: blue; text-decoration: underline; ">kirkmcd@Princeton.EDU</a>><br><br><br><br><o:p></o:p></div><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Folks,<o:p></o:p></div><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">There is a technical question as to how we should be calculating<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">emittance for beams in electromagnetic fields.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">The formal theory of Liouville’s theorem is clear that the invariant<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">volume in phase space is to be calculated with the canonical momentum<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">gamma m v + e A / c<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">and not the mechanical momentum m v.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">This is awkward in two ways:<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">1. We don’t always know the vector potential of our fields<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">2. The vector potential is subject to gauge transformations, so<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">canonical momentum is not gauge invariant.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">The second issue is disconcerting in that it suggests that phase-space<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">volume, and emittance, are not actually invariant -- with respect to<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">gauge transformations.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Hence, it is useful to note a very old paper,<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">W.F.G. Swann, Phys. Rev. 44, 233 (1933)<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">which shows that the phase-space volume for a set of noninteracting<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">particles is the same whether or not the term e A / c is included in<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">the “momentum”.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">This result has the consequence that phase-space volume (and<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">emittance) is actually gauge invariant – although the location of a<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">volume element in space space is gauge dependent.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">---------------<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">This suggests that we could simply calculate emittances based only on<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">the mechanical momentum, and avoid having to worry about the accuracy<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">of our model for the vector potential.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Of course, our calculations are actually of rms emittance, which is a<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">better representation of the “ideal” emittance if the phase-space<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">volume is more “spherical”, and not elongated/twisted.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">It could be that the shape of the phase-space volume is better for rms<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">emittance calculation if the vector potential, in some favored gauge,<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">is included in the calculation.....<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">--Kirk<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">PS I have placed Swann’s paper as DocDB 560<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><a href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase" style="color: blue; text-decoration: underline; ">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</a><o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">user = ionization pass = mucollider1<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">See also the paper by Lemaitre that used Liouville’s theorem for<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">cosmic rays in the Earth’s atmosphere (using mechanical momentum). <o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">This may well be the earliest paper about particle beams and<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Liouville’s theorem.<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">PPS Scott Berg notes that when one evaluates emittance at a fixed<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">plane in space, rather than at a fixed time, it is better to use the<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">“longitudinal” coordinates (E,t) rather than (P_z,z).<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Is there any written reference that explains this “well known” fact?<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">How is this prescription affected by electromagnetic fields?<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">The vector potential of even a simple rf accelerating cavity has an<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">A_z component (which is zero on axis, but nonzero off it).<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><a href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf" style="color: blue; text-decoration: underline; ">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</a><o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Note that the vector potential is nonzero outside the cavity, even<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">though the E and B fields are zero there!<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><o:p> </o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">Do we know how to include A_z in our longitudinal emittance calculations?<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">_______________________________________________<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; ">MAP-l mailing list<o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><a href="mailto:MAP-l@lists.bnl.gov" style="color: blue; text-decoration: underline; ">MAP-l@lists.bnl.gov</a><o:p></o:p></div></blockquote><blockquote style="margin-top: 5pt; margin-bottom: 5pt; "><div style="margin-top: 0in; margin-right: 0in; margin-bottom: 0.0001pt; margin-left: 0in; font-size: 12pt; font-family: 'Times New Roman', serif; "><a href="https://lists.bnl.gov/mailman/listinfo/map-l" style="color: blue; text-decoration: underline; ">https://lists.bnl.gov/mailman/listinfo/map-l</a><o:p></o:p></div></blockquote><div style="margin-top: 0in; margin-right: 0in; 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