<HTML><HEAD></HEAD> <BODY style="WORD-WRAP: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space" dir=ltr> <DIV dir=ltr> <DIV style="FONT-FAMILY: 'Arial'; COLOR: #000000; FONT-SIZE: 10pt"> <DIV>Alex,</DIV> <DIV> </DIV> <DIV>Your latest email addresses mainly the important issue of whether we can/should be using some other form of approximation to emittance calculations than we are presently using.</DIV> <DIV> </DIV> <DIV>I infer that the answer is NO, but that we should be aware of limitations of the scheme we do use.</DIV> <DIV> </DIV> <DIV>Question: When one uses “rms emittance” calculations, are the results dependent on the choice of gauge, or whether or not we include the vector potential at all?</DIV> <DIV> </DIV> <DIV>The vector potential is nonzero outside a closed fry cavity, but the cavity has no effect on the motion of particles when they are outside it.  If the “rms emittance” for the particles outside the cavity depends on the vector potential outside the cavity, that will continue to disconcert me.</DIV> <DIV> </DIV> <DIV>Alexander argued, I believe, that in case the magnetic field is uniform and axial, the rms emittance is independent of whether or not the vector potential is included (which implies that this emittance is gauge invariant).</DIV> <DIV> </DIV> <DIV>Does this happy result extend to arbitrary electromagnetic fields, as is the case of phase volume?</DIV> <DIV> </DIV> <DIV><FONT face="Times New Roman"><FONT style="FONT-SIZE: 10.8pt">You say: “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.”</FONT></FONT></DIV> <DIV><FONT face="Times New Roman"></FONT> </DIV> <DIV><FONT face="Times New Roman">Can I show my ignorance by asking what is a “linear map”?  I have the impression that transport through field-free vacuum is NOT an example of a “linear map”.</FONT></DIV> <DIV> </DIV> <DIV>--Kirk</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> </DIV> <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=kirkmcd@Princeton.EDU href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</A> ; <A title=map-l@lists.bnl.gov href="mailto:map-l@lists.bnl.gov">MAP List</A> ; <A title=rdryne@lbl.gov href="mailto:rdryne@lbl.gov">Robert D Ryne</A> ; <A title=dragtnb@comcast.net href="mailto:dragtnb@comcast.net">alex dragt</A> ; <A title=srmane001@gmail.com href="mailto:srmane001@gmail.com">srmane001@gmail.com</A> </DIV> <DIV><B>Subject:</B> 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></DIV></DIV></DIV></BODY></HTML>