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<DIV>Folks,</DIV>
<DIV> </DIV>
<DIV>This email greatly clarifies things for me, thanks to help from Alex
Dragt.</DIV>
<DIV> </DIV>
<DIV>One aspect of this clarification is that we may be making a formal error in
the way we calculate (longitudinal) emittance when fields are present.</DIV>
<DIV> </DIV>
<DIV>--Kirk</DIV>
<DIV> </DIV>
<DIV>This email takes up the related theme that we prefer to use (x,y,t), rather
than (x,y,z) as the q coordinates, with z rather than t as the independent
variable in the Hamiltonian.</DIV>
<DIV> </DIV>
<DIV>Thanks to Alex for pointing me to sec.1.6 of his impressive e-book!</DIV><A
href="http://www.physics.umd.edu/dsat/"><FONT face="Times New Roman"><FONT
style="FONT-SIZE: 12pt">http://www.physics.umd.edu/dsat/</FONT></FONT></A><BR>
<DIV> </DIV>
<DIV>What I get from this is that the canonical momentum associated with
coordinate t is</DIV>
<DIV>p_t = – H, where H = the Hamiltonian.</DIV>
<DIV> </DIV>
<DIV>For a charged particle in an external electromagnetic field, the
Hamiltonian H includes both the scalar potential V and the vector potential
A.</DIV>
<DIV> </DIV>
<DIV>[Dragt gives the Hamiltonian using coordinates (x,y,z) in his eq. (1.5.130)
on .pdf page 142 = text page 88, and the Hamiltonian using coordinates (x,y,t)
in his eq. (1.6.16) on .pdf page 153 = text page 99.]</DIV>
<DIV> </DIV>
<DIV>The Hamiltonian is NOT the mechanical energy when nonzero electromagnetic
fields are present.</DIV>
<DIV> </DIV>
<DIV>So, when calculating “emittances” in these coordinates, we should NOT be
using the energy E, but the Hamiltonian H as the “momentum” associated with
coordinate t.</DIV>
<DIV> </DIV>
<DIV>This is contrary to what I have been told we are doing in, say, the
programs ICOOL/ECALC, although perhaps I have a misunderstanding about
this.</DIV>
<DIV>I have the impression that we are using E rather than H, but that we do use
canonical momenta in the x and y coordinates when calculating emittances.</DIV>
<DIV>This might be the source of some numerical troubles that may exist in
recent calculations.</DIV>
<DIV> </DIV>
<DIV>I presume that in the Hamiltonian formalism with coordinates (x,y,t) and
canonical momenta </DIV>
<DIV>(p_mech,x + q A_x / c,p_mech,y + q A_y /c,-H)</DIV>
<DIV>that the density in this space space of a set of noninteracting particles
is invariant with respect to the independent variable z, according to
Liouville’s theorem.</DIV>
<DIV> </DIV>
<DIV>Next, I review Swann’s argument to see what happens if we simply ignore the
scalar and vector potential. In this case we would associate the
mechanical energy E as the (noncanonical) “momentum” of coordinate t. So,
the goal is to keep using coordinates (x,y,t) but to switch to “momenta”
(p_mech,x,p_mech,y,E).</DIV>
<DIV> </DIV>
<DIV>The off-diagonal elements in the Jacobian (Swann, p. 226) again don’t
matter.</DIV>
<DIV>However, we have a problem in that the (E,H) diagonal matrix element
is</DIV>
<DIV>partial H / partial E, which is not equal to 1 when fields are
present.</DIV>
<DIV> </DIV>
<DIV>This means that we cannot simply ignore the vector (and scalar) potentials
when calculating phase volumes (and presumably also emittance) when using
coordinates (x,y,t) rather than (x,y,z).</DIV>
<DIV> </DIV>
<DIV>I believe that a gauge transformation remains a canonical transformation
when using the coordinates (x,y,t), so phase volume (and presumably also
emittance) is gauge invariant.</DIV>
<DIV> </DIV>
<DIV>Since we are strongly committed to the use of coordinates (x,y,t), it seems
that we will have to continue including the vector (and scalar!) potentials in
our emittance calculations.</DIV>
<DIV> </DIV>
<DIV>We may presently be making an error in using E_mech, rather than H as the
canonical momentum associated with coordinate t.</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV> </DIV>
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<DIV style="BACKGROUND: #f5f5f5">
<DIV style="font-color: black"><B>From:</B> <A title=dragtnb@comcast.net
href="mailto:dragtnb@comcast.net">alex dragt</A> </DIV>
<DIV><B>Sent:</B> Friday, March 11, 2011 4:24 PM</DIV>
<DIV><B>To:</B> <A title=nsergei@fnal.gov href="mailto:nsergei@fnal.gov">Sergei
Nagaitsev</A> ; <A title=summers@phy.olemiss.edu
href="mailto:summers@phy.olemiss.edu">Don Summers</A> ; <A
title=alexahin@fnal.gov href="mailto:alexahin@fnal.gov">Yuri Alexahin</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=kirkmcd@Princeton.EDU href="mailto:kirkmcd@Princeton.EDU">Kirk T
McDonald</A> ; <A title=srmane001@gmail.com
href="mailto:srmane001@gmail.com">srmane001@gmail.com</A> </DIV>
<DIV><B>Subject:</B> [MAP] Liouville's theorem, kinematic invariants, and
dynamic invariants</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"><FONT
class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span>Dear All,</SPAN></FONT>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> Confusion reigns:
</SPAN></FONT><SPAN style="FONT-SIZE: 18px" class=Apple-style-span>As described
in the papers</SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span><BR></SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">A. Dragt, F. Neri, et al., "LIE
ALGEBRAIC TREATMENT OF LINEAR AND NONLINEAR BEAM DYNAMICS",</DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">Annual Review of Nuclear and
Particle Science 38, p. 455 (1988).</DIV></SPAN></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">A. Dragt, R. Gluckstern, et al.,
"THEORY OF EMITTANCE INVARIANTS", Lecture Notes in</DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">Physics 343: Proceedings of the
Joint US-CERN Capri School on Accelerator Physics,</DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">Springer Verlag (1989).</DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica"> </DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">
<DIV style="MARGIN: 0px; FONT: 12px helvetica">A. Dragt, F. Neri, et al.,
"GENERAL MOMENT INVARIANTS FOR LINEAR HAMILTONIAN SYSTEMS", </DIV>
<DIV style="MARGIN: 0px; FONT: 12px helvetica">Physical Review A, 45, p. 2572
(1992).</DIV>
<DIV
style="MARGIN: 0px; FONT: 12px helvetica"> </DIV></DIV></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span>there are two kinds of invariants, which I call
"kinematic" and "dynamic". It is important not to confound
them.</SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> The Courant-Snyder invariants,
generally treated in the linear approximation as described by Sergei
below, are examples of dynamic invariants. Their use requires a
knowledge of the equations of motion or, equivalently, a one-period or one-turn
map. (They generally apply to periodic systems for which the linear part
of the map has all its eigenvalues on the unit circle and distinct, but in
principle could also be constructed when the eigenvalues are off the unit
circle.) Contrary to the reservations that appear to be expressed by
Sergei, they are extendable to the nonlinear case using normal form
methods. See, for example, the first paper above and Sections 8.10 and
8.11 of the MaryLie manual available at the Web site</SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><A
href="http://www.physics.umd.edu/dsat/">http://www.physics.umd.edu/dsat/</A></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span>There are also papers in the literature that extend them,
at least in the linear case, to non-periodic systems.</SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> Examples of kinematic invariants
include what I call "moment invariants" and "eigen emitances". They
involve moments of particle distributions and, like Liouville's theorem, make no
specific use of the equations of motion save for the symplectic condition.
It is these invariants that are the current focus of interest with regard to
questions of "emittance partitioning". See the draft
paper</SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><SPAN
style="FONT-FAMILY: 'Lucida Grande', helvetica, arial, verdana, sans-serif; FONT-SIZE: 13px"
class=Apple-style-span>
<TABLE style="Z-INDEX: auto; POSITION: static; COLOR: #000000"
summary="Additional metadata">
<TBODY>
<TR>
<TD
style="PADDING-BOTTOM: 0em; PADDING-LEFT: 0em; PADDING-RIGHT: 0.5em; VERTICAL-ALIGN: top; FONT-WEIGHT: bold; PADDING-TOP: 0.1em"
class="tablecell arxivid"><A style="TEXT-DECORATION: none"
href="http://arxiv.org/abs/1010.1558v2">arXiv:1010.1558v2</A>
[physics.acc-ph]</TD></TR></TBODY></TABLE><BR></SPAN></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><SPAN
style="FONT-FAMILY: 'Lucida Grande', helvetica, arial, verdana, sans-serif; FONT-SIZE: 13px"
class=Apple-style-span><SPAN style="FONT-FAMILY: helvetica; FONT-SIZE: 18px"
class=Apple-style-span>Their computation and use are described in the papers
above and Section 8.37 of the MaryLie
manual.</SPAN> </SPAN></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> Moment invariants are currently
known only for the case of linear maps, but there is some evidence that they
should also exist in the nonlinear case. See Chapter 26 (only
partially complete) and </SPAN></FONT><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span>Section 6.8.2</SPAN><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> </SPAN><FONT class=Apple-style-span size=5><SPAN
style="FONT-SIZE: 18px" class=Apple-style-span>of "Lie Methods ...", also
available at the Web site above. </SPAN></FONT><SPAN
style="FONT-SIZE: 18px" class=Apple-style-span>In the past I have tried to
construct moment invariants for nonlinear maps by finding Casimir operators for
the full Lie algebra of all symplectic maps. The problem is difficult
because the usual method for constructing Casimir operators requires that the
Killing metric be invertible. See Section 21.11 of "Lie Methods
...". This metric is not invertible in the nonlinear case, and therefore
the standard machinery for constructing Casimirs fails. With considerable
effort I found an alternate approach with some promising results, but was
eventually overwhelmed by algebraic complexity. Moment invariants for the
nonlinear case, if they exist at all, will require many pages even to write them
down.</SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span><BR></SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span> I hope these remarks provide
some clarification.</SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span><BR></SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span>Best,</SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span><BR></SPAN></DIV>
<DIV><SPAN style="FONT-SIZE: 18px" class=Apple-style-span>Alex</SPAN></DIV>
<DIV><FONT class=Apple-style-span size=5><SPAN style="FONT-SIZE: 18px"
class=Apple-style-span><BR></SPAN></FONT></DIV>
<DIV>
<DIV>
<DIV>On Mar 10, 2011, at 7:32 PM, Sergei Nagaitsev wrote:</DIV><BR
class=Apple-interchange-newline>
<BLOCKQUOTE type="cite">
<DIV>Dear Kirk,<BR><BR>let me add my two cents to this
discussion:<BR><BR>1. For the beam emittance to be a useful quantity, it
needs to be conserved as the beam propagates along the beamline (consider
non-interacting particles) through various external electromagnetic
fields. To this end, the definition of emittance that reflects such a
property is "an ensemble-averaged action". The particle motion is
assumed to be integrable, i.e. there exists 3 functionally-independent
constants of the motion in involution with the Hamiltonian. Particle
actions, expressed through such constants of motion, are also constants of
motion. Thus, the average actions (emittances) are conserved. In a
linear-focusing transport channel with linear rf focusing, such constants of
motion exist and are called Courant-Snyder invariants (2 transverse and 1
longitudinal). In a nonlinear (and generally time-dependent) focusing
channel such constants of motion might not exist, therefore, the emittance (as
a conserved quantity) is not defined. We may use an approximate
expression for the emittance by using linear-only focusing to define actions
and then by treating non-linearities as a perturbation. This leads to an
effective emittance growth if a beamline has nonlinear elements even though
the Liouville's theorem states that the phase-space density is
conserved. In some cases this emittance growth is not "real"
(irreversible) but just a reflection that we are using an incorrect action
definition. Finally, my definition of the emittance (as the average
action) is identical to the definition through eigen-values of a sigma matrix
only in a case of a linear focusing channel. However, where possible
(like in case of a bunch occupying a large portion of an rf bucket) we should
use exact actions, not approximate. <BR><BR>2. When averaging particle actions
over the distribution function at a given time t it is useful to remember that
a time slice t=const in one frame is not t'=const in another frame because of
Lorentz transformations.<BR><BR>Sergei <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 8:49 pm<BR>Subject: Re: [MAP] Liouville's theorem and
electromagnetic fields<BR>To: alex dragt <<A
href="mailto:dragtnb@comcast.net">dragtnb@comcast.net</A>><BR>Cc: MAP List
<<A
href="mailto:map-l@lists.bnl.gov">map-l@lists.bnl.gov</A>><BR><BR><BR>
<BLOCKQUOTE type="cite">Folks,<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">I have added Alex' paper to DocDB 560. See
Appendix A.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">It is gratifying to see that the fact that
Liouville's theorem holds <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">for <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">both mechanical and canonical phase space is "well
known to those who <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">know".<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">The challenge now is to learn how best to use the
"freedom" offered to <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">us by <BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">this apparently nonintuitive result.<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite">--Kirk<BR></BLOCKQUOTE>
<BLOCKQUOTE type="cite"><BR></BLOCKQUOTE>
<BLOCKQUOTE
type="cite">--------------------------------------------------<BR></BLOCKQUOTE><BR></DIV></BLOCKQUOTE></DIV>
<DIV> </DIV></DIV>
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