<HTML><HEAD> <META content="text/html; charset=UTF-8" http-equiv=Content-Type></HEAD> <BODY dir=ltr bgColor=#ffffff text=#000000> <DIV dir=ltr> <DIV style="FONT-FAMILY: 'Arial'; COLOR: #000000; FONT-SIZE: 10pt"> <DIV>Alexey,</DIV> <DIV> </DIV> <DIV>For the subspace (q,p) we have</DIV> <DIV> </DIV> <DIV>dq’ dp’ = J dq dp</DIV> <DIV> </DIV> <DIV>J = | dq’/dq  dq’/dp |</DIV> <DIV>      | dp’/dq  dp’/dp |</DIV> <DIV> </DIV> <DIV>Suppose p = m v + A   (in units where e/c = 1)</DIV> <DIV>and we transform</DIV> <DIV>q’ = q</DIV> <DIV>p’ = mv = p – A(q)</DIV> <DIV> </DIV> <DIV>Then the Jacobian is</DIV> <DIV>J = |       1     0 |</DIV> <DIV>      | –dA/dq  1 | = 1</DIV> <DIV> </DIV> <DIV>It looks to me like the partial phase volumes are also invariant under the “transformation” of neglecting the vector potential.</DIV> <DIV> </DIV> <DIV>--Kirk</DIV> <DIV> </DIV> <DIV style="FONT-STYLE: normal; DISPLAY: inline; FONT-FAMILY: 'Calibri'; COLOR: #000000; FONT-SIZE: small; FONT-WEIGHT: normal; TEXT-DECORATION: none"> <DIV style="FONT: 10pt tahoma"> <DIV><FONT face=Arial></FONT> </DIV> <DIV style="BACKGROUND: #f5f5f5"> <DIV style="font-color: black"><B>From:</B> <A title=burov@fnal.gov href="mailto:burov@fnal.gov">Alexey Burov</A> </DIV> <DIV><B>Sent:</B> Thursday, March 10, 2011 5:33 PM</DIV> <DIV><B>To:</B> <A title=map-l@lists.bnl.gov href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</A> </DIV> <DIV><B>Subject:</B> Re: [MAP] Liouville's theorem and electromagnetic fields</DIV></DIV></DIV> <DIV> </DIV></DIV> <DIV style="FONT-STYLE: normal; DISPLAY: inline; FONT-FAMILY: 'Calibri'; COLOR: #000000; FONT-SIZE: small; FONT-WEIGHT: normal; TEXT-DECORATION: none">One remark to Swann's paper: <BR>His theorem relates to the total emittance, not to the partial ones. Partial emittances are sensitive to eA/c term. <BR><BR>A possible way to get rid of eA/c inside solenoidal structures is to make a fake 0-length edge of the solenoid at a place where emittances are calculated; kicks from the edge solenoidal fields have to be taken into account, of course. <BR><BR>Alexey. <BR><BR>On 3/10/2011 4:09 PM, Kirk T McDonald wrote: <BLOCKQUOTE cite=mid:468B48A3C96B4BA3AA66387F9E650168@mumu30 type="cite"> <DIV dir=ltr> <DIV style="FONT-FAMILY: 'Arial'; COLOR: rgb(0,0,0); FONT-SIZE: 10pt"> <DIV>Folks,</DIV> <DIV> </DIV> <DIV>There is a technical question as to how we should be calculating emittance for beams in electromagnetic fields.</DIV> <DIV> </DIV> <DIV>The formal theory of Liouville’s theorem is clear that the invariant volume in phase space is to be calculated with the canonical momentum</DIV> <DIV>gamma m v + e A / c</DIV> <DIV>and not the mechanical momentum m v.</DIV> <DIV> </DIV> <DIV>This is awkward in two ways:</DIV> <DIV>1.   We don’t always know the vector potential of our fields</DIV> <DIV>2.   The vector potential is subject to gauge transformations, so canonical momentum is not gauge invariant.</DIV> <DIV> </DIV> <DIV>The second issue is disconcerting in that it suggests that phase-space volume, and emittance, are not actually invariant  -- with respect to gauge transformations.</DIV> <DIV> </DIV> <DIV>Hence, it is useful to note a very old paper,</DIV> <DIV>W.F.G. Swann, Phys. Rev. 44, 233 (1933)</DIV> <DIV>which shows that the phase-space volume for a set of noninteracting particles is the same whether or not the term e A / c is included in the “momentum”.</DIV> <DIV> </DIV> <DIV>This result has the consequence that phase-space volume (and emittance) is actually gauge invariant – although the location of a volume element in space space is gauge dependent.</DIV> <DIV> </DIV> <DIV>---------------</DIV> <DIV>This suggests that we could simply calculate emittances based only on the mechanical momentum, and avoid having to worry about the accuracy of our model for the vector potential.</DIV> <DIV> </DIV> <DIV>Of course, our calculations are actually of rms emittance, which is a better representation of the “ideal” emittance if the phase-space volume is more “spherical”, and not elongated/twisted.</DIV> <DIV> </DIV> <DIV>It could be that the shape of the phase-space volume is better for rms emittance calculation if the vector potential, in some favored gauge, is included in the calculation.....</DIV> <DIV> </DIV> <DIV>--Kirk</DIV> <DIV> </DIV> <DIV>PS  I have placed Swann’s paper as DocDB 560</DIV> <DIV><A title=http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase" moz-do-not-send="true">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</A></DIV> <DIV>user = ionization pass = mucollider1</DIV> <DIV> </DIV> <DIV>See also the paper by Lemaitre that used Liouville’s theorem for cosmic rays in the Earth’s atmosphere (using mechanical momentum).   This may well be the earliest paper about particle beams and Liouville’s theorem.</DIV> <DIV> </DIV> <DIV>PPS  Scott Berg notes that when one evaluates emittance at a fixed plane in space, rather than at a fixed time, it is better to use the “longitudinal” coordinates (E,t) rather than (P_z,z).</DIV> <DIV> </DIV> <DIV>Is there any written reference that explains this “well known” fact?</DIV> <DIV> </DIV> <DIV>How is this prescription affected by electromagnetic fields?</DIV> <DIV> </DIV> <DIV>The vector potential of even a simple rf accelerating cavity has an A_z component (which is zero on axis, but nonzero off it).</DIV> <DIV><A title=http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf href="http://puhep1.princeton.edu/%7Emcdonald/examples/cylindrical.pdf" moz-do-not-send="true">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</A></DIV> <DIV>Note that the vector potential is nonzero outside the cavity, even though the E and B fields are zero there!</DIV> <DIV> </DIV> <DIV>Do we know how to include A_z in our longitudinal emittance calculations?</DIV></DIV></DIV><PRE wrap=""><FIELDSET class=mimeAttachmentHeader></FIELDSET> _______________________________________________ MAP-l mailing list <A class=moz-txt-link-abbreviated href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</A> <A class=moz-txt-link-freetext href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</A> </PRE></BLOCKQUOTE><BR> <P> <HR> _______________________________________________<BR>MAP-l mailing list<BR>MAP-l@lists.bnl.gov<BR>https://lists.bnl.gov/mailman/listinfo/map-l<BR></DIV></DIV></DIV></BODY></HTML>