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<DIV><FONT size=2 face=Arial>
<DIV><FONT size=2 face=Arial>Rob,</FONT></DIV>
<DIV><FONT size=2 face=Arial></FONT> </DIV>
<DIV><FONT size=2 face=Arial>1. <FONT face="Times New Roman">"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)."</FONT></FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>This kind of thinking is what follows from emphasis on the
canonical/symplectic point of view.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>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.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>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.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>Nonetheless, it would be a service to mankind if this insight
could be documented in a manner that ordinary mortals can
understand.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>I note that you evade the important question of how this works
in the presence of electromagnetic fields.</FONT></DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT size=2>--Kirk</FONT></DIV></FONT></DIV>
<DIV style="FONT: 10pt Tahoma">
<DIV><BR></DIV>
<DIV style="BACKGROUND: #f5f5f5">
<DIV style="font-color: black"><B>From:</B> <A title=rdryne@lbl.gov
href="mailto:rdryne@lbl.gov">Robert D Ryne</A> </DIV>
<DIV><B>Sent:</B> Thursday, March 10, 2011 7:33 PM</DIV>
<DIV><B>To:</B> <A title=alexahin@fnal.gov href="mailto:alexahin@fnal.gov">Yuri
Alexahin</A> ; <A title=kirkmcd@Princeton.EDU
href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</A> </DIV>
<DIV><B>Cc:</B> <A title=map-l@lists.bnl.gov
href="mailto:map-l@lists.bnl.gov">MAP List</A> ; <A title=dragtnb@comcast.net
href="mailto:dragtnb@comcast.net">alex dragt</A> ; <A title=dragtg5@comcast.net
href="mailto:dragtg5@comcast.net">Alex Dragt</A> ; <A
title=dragt@physics.umd.edu href="mailto:dragt@physics.umd.edu">Alex Dragt</A>
</DIV>
<DIV><B>Subject:</B> Re: [MAP] Liouville's theorem and electromagnetic
fields</DIV></DIV></DIV>
<DIV><BR></DIV><SPAN style="FONT-SIZE: 17px" class=Apple-style-span>I have not
yet read the papers mentioned. But here are some brief comments. Alex Dragt and
I (cc to Alex) have been thinking about this a lot in the past months.</SPAN>
<DIV style="FONT-SIZE: 17px"><BR></DIV>
<DIV style="FONT-SIZE: 17px">The natural quantities to be computed are called
"eigen-emittances."</DIV>
<DIV style="FONT-SIZE: 17px">To compute them properly they need to be derived
from a beam 2nd moment matrix, Sigma, formed using canonical variables.</DIV>
<DIV style="FONT-SIZE: 17px">The eigen-emittances are invariant under linear
symplectic transformations.</DIV>
<DIV style="FONT-SIZE: 17px"><BR></DIV>
<DIV style="FONT-SIZE: 17px">The eigen-emittances can be computed in various
ways, but the simplest is to compute the eigen-values of J Sigma, where J is the
fundamental symplectic 2-form; the eigen-emittances are the modulii of the
eigen-values of J Sigma (which are pure imaginary and in +/- pairs). If one is
interested in calculating the symplectic matrix that transforms Sigma to
Williamson normal form, Alex Dragt has an algorithm to do this and has
implemented it in the MaryLie code.</DIV>
<DIV style="FONT-SIZE: 17px"><BR></DIV>
<DIV style="FONT-SIZE: 17px">Though the entries of Sigma will depend on the
choice of gauge, the eigen-emittances themselves are gauge invariant. We can't
just set the vector potential to zero inside elements where it is nonzero, and
expect to calculate the correct eigen-emittances (as was suggested below).</DIV>
<DIV style="FONT-SIZE: 17px"><BR></DIV>
<DIV style="FONT-SIZE: 17px">
<BLOCKQUOTE style="FONT-SIZE: medium" type="cite">
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<BLOCKQUOTE cite=mid:468B48A3C96B4BA3AA66387F9E650168@mumu30
type="cite"><DIV dir=ltr>
<DIV style="FONT-FAMILY: Arial; COLOR: rgb(0,0,0); FONT-SIZE: 10pt">
<DIV>PPS Scott Berg notes that when one evaluates emittance at a
fixed plane in space, rather than at a fixed time, it is better to use
the “longitudinal” coordinates (E,t) rather than (P_z,z).</DIV>
<DIV> </DIV>
<DIV>Is there any written reference that explains this “well known”
fact?</DIV>
<DIV> </DIV></DIV></DIV></BLOCKQUOTE></DIV></DIV></DIV></BLOCKQUOTE></DIV></DIV></DIV></BLOCKQUOTE></DIV></BLOCKQUOTE></DIV>
<DIV style="FONT-SIZE: 17px"> </DIV>
<DIV style="FONT-SIZE: 17px">The above follows directly from whether we use the
time t as the independent variable or the Cartesian coordinate z as the
independent variable. When using the time, the longitudinal variables are
(z,p_{z,canonical}). When using z, the longitudinal variables are (t, -E) where
t is arrival time at location z, and where E is the total energy of a particle
when it reaches location z, i.e. E=\gamma m c^2 + q \Phi.</DIV>
<DIV style="FONT-SIZE: 17px"><BR></DIV>
<DIV style="FONT-SIZE: 17px">Rob</DIV>
<DIV><BR></DIV>
<DIV>
<DIV>On Mar 10, 2011, at 4:29 PM, Yuri Alexahin wrote:</DIV><BR
class=Apple-interchange-newline>
<BLOCKQUOTE type="cite">
<DIV>Hi Kirk,<BR><BR>Thank you for digging out these interesting papers.<BR>Of
course the Poincare invariants remain the same no matter what momenta are
used.<BR>But this is not what we calculate from tracking or measurement data
using standard definition.<BR>So a clarification is still needed of what and
how we should calculate.<BR><BR>Yuri<BR><BR>----- Original Message
-----<BR>From: Kirk T McDonald <<A
href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</A>><BR>Date:
Thursday, March 10, 2011 4:09 pm<BR>Subject: [MAP] Liouville's theorem and
electromagnetic fields<BR>To: MAP List <<A
href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</A>><BR>Cc: Kirk
McDonald <<A
href="mailto:kirkmcd@Princeton.EDU">kirkmcd@Princeton.EDU</A>><BR><BR><BR>
<BLOCKQUOTE type="cite">Folks,<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">There is a technical question as to how we should be
calculating <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">emittance for beams in electromagnetic
fields.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">The formal theory of Liouville’s theorem is clear
that the invariant <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">volume in phase space is to be calculated with the
canonical momentum<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">gamma m v + e A / c<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">and not the mechanical momentum m v.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">This is awkward in two ways:<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">1. We don’t always know the vector
potential of our fields<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">2. The vector potential is subject to
gauge transformations, so <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">canonical momentum is not gauge
invariant.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">The second issue is disconcerting in that it
suggests that phase-space <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">volume, and emittance, are not actually invariant
-- with respect to <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">gauge transformations.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Hence, it is useful to note a very old
paper,<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">W.F.G. Swann, Phys. Rev. 44, 233
(1933)<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">which shows that the phase-space volume for a set of
noninteracting <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">particles is the same whether or not the term e A /
c is included in <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">the “momentum”.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">This result has the consequence that phase-space
volume (and <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">emittance) is actually gauge invariant – although
the location of a <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">volume element in space space is gauge
dependent.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">---------------<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">This suggests that we could simply calculate
emittances based only on <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">the mechanical momentum, and avoid having to worry
about the accuracy <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">of our model for the vector
potential.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Of course, our calculations are actually of rms
emittance, which is a <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">better representation of the “ideal” emittance if
the phase-space <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">volume is more “spherical”, and not
elongated/twisted.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">It could be that the shape of the phase-space volume
is better for rms <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">emittance calculation if the vector potential, in
some favored gauge, <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">is included in the calculation.....<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">--Kirk<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">PS I have placed Swann’s paper as DocDB
560<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><A
href="http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase">http://nfmcc-docdb.fnal.gov:8080/cgi-bin/DocumentDatabase</A><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">user = ionization pass = mucollider1<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">See also the paper by Lemaitre that used Liouville’s
theorem for <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">cosmic rays in the Earth’s atmosphere (using
mechanical momentum). <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">This may well be the earliest paper about particle
beams and <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Liouville’s theorem.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">PPS Scott Berg notes that when one evaluates
emittance at a fixed <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">plane in space, rather than at a fixed time, it is
better to use the <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">“longitudinal” coordinates (E,t) rather than
(P_z,z).<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Is there any written reference that explains this
“well known” fact?<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">How is this prescription affected by electromagnetic
fields?<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">The vector potential of even a simple rf
accelerating cavity has an <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">A_z component (which is zero on axis, but nonzero
off it).<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><A
href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</A><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Note that the vector potential is nonzero outside
the cavity, even <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">though the E and B fields are zero
there!<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">Do we know how to include A_z in our longitudinal
emittance calculations?<BR></BLOCKQUOTE>
<BLOCKQUOTE
type="cite">_______________________________________________<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">MAP-l mailing list<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><A
href="mailto:MAP-l@lists.bnl.gov">MAP-l@lists.bnl.gov</A><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><A
href="https://lists.bnl.gov/mailman/listinfo/map-l">https://lists.bnl.gov/mailman/listinfo/map-l</A><BR></BLOCKQUOTE>_______________________________________________<BR>MAP-l
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