<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta content="text/html; charset=ISO-8859-1" http-equiv="Content-Type"> </head> <body bgcolor="#ffffff" text="#000000"> <font face="Arial" size="2"></font><font face="Arial" size="2">Kirk, <br> sorry, but your statement below referring my name is not correct:<br> <br> "Give versions of Swann's theorem for rms emittance calculations.  Clarify whether the theorem holds only for 6D emittance, or for 4D and 2D subemittances as well.</font> <div><font face="Arial" size="2">Alexander Shemyakin claims that this has been done for the 4D transverse rms emittance for beams inside an ideal DC solenoids, but perhaps not for general electromagnetic fields.   <i>Apparently, Alexey Burov does not believe the theorem holds even for this special case."<br> <br> </i>As I wrote in my very first message to you, the partial emittances, calculated with mechanical sigma-matrix, are not correct. But their product, the total emittance, is correct. For the e-beam inside the solenoid, the longitudinal emittance is decoupled, so the total emittance is 4D. Thus, 4D mechanical emittance is the same as canonical. It is clearly written in the Lebedev-Bogacz paper, in particular. A. Shemyakin also correctly mentioned, that there is no disagreemnt between him and me.<br> <br> Alexey.<br> </font></div> <br> On 3/12/11 8:03 PM, Kirk T McDonald wrote: <blockquote cite="mid:CBCA4B1192234A658030B6355F2C710D@mumu24" type="cite"> <meta content="text/html; charset=ISO-8859-1" http-equiv="Content-Type"> <meta name="GENERATOR" content="MSHTML 8.00.6001.19019"> <div><font face="Arial" size="2">Alex,</font></div> <div> </div> <div><font face="Arial" size="2">Thanks for that useful clarification.  I now agree that Rob's slide 16 is correct.</font></div> <div> </div> <div><font face="Arial" size="2">This leads to several other comments.</font></div> <div> </div> <div><font face="Arial" size="2">If you find this tedious, but are willing to consider some of the argument, skip to item 7.</font></div> <div> </div> <div><font face="Arial" size="2">1.  With H = E_mech + q V,</font></div> <div><font face="Arial" size="2">the partial derivative on the diagonal of the Jacobian of the transformation from </font></div> <div><font face="Arial" size="2">canonical coordinates</font></div> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y_qA_y/c, H=E_mech+qV)</font></div> <div><font face="Arial" size="2">to (noncanonical) coordinates</font></div> <div><font face="Arial" size="2">(x, y, t, p_mech,x, p_mech,y, E_mech)</font></div> <div><font face="Arial" size="2">is</font></div> <div><font face="Arial" size="2">partial H / partial E_mech = 1.</font></div> <div> </div> <div><font face="Arial" size="2">Hence, the determinant of the Jacobian is 1, which means that the phase volume is the same whether or not one includes the potentials A and V in the calculation, using either t or z as the independent variable.   This a the generalization of Swann's "theorem".</font></div> <div> </div> <div><font face="Arial" size="2">So, in principle, we are free to leave out all consideration of the potentials in calculation of phase volume, and its approximate representation by emittance!</font></div> <div> </div> <div><font face="Arial" size="2">There remains the vexing issue of which potentials, I.e., which gauge, to use such that the numerical value of the calculated emittance is the best representation of the underlying phase volume.</font></div> <div> </div> <div><font face="Arial" size="2">2.  If we choose to include the potentials A and V in the calculation, we must do so consistently.</font></div> <div> </div> <div><font face="Arial" size="2">However, we can give a corollary to Swann's theorem which indicates that in principle we have a lot of flexibility.</font></div> <div> </div> <div><font face="Arial" size="2">For example, if we decide to change from the canonical coordinates</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, H=E_mech+qV)</font></div> <div><font face="Arial" size="2">to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)</font></div> <div><font face="Arial" size="2">or to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y, H=E_mech+qV)</font></div> <div><font face="Arial" size="2">or to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x, p_mech_y+qA_y/c, H=E_mech+qV)</font></div> <div><font face="Arial" size="2">or to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y, E_mech)</font></div> <div><font face="Arial" size="2">or to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x, p_mech_y+qA_y/c, E_mech)</font></div> <div><font face="Arial" size="2">or to</font></div> <div> <div><font face="Arial" size="2">(x, y, t, p_mech_x, p_mech_y, H=E_mech+qV),</font></div> <div><font face="Arial" size="2">in every case the determinant of the Jacobian is 1, so phase volume is invariant under all 6 of these transformations [as well as under the original transformation to coordinates</font></div> <div><font face="Arial" size="2">(x, y, t, p_mech,x, p_mech,y, E_mech) ].</font></div> <div> </div> <div><font face="Arial" size="2">[Similarly, if we use t as the independent variable, we obtain 7 versions of Swann's theorem, for a total of 14 versions so far.]</font></div> <div> </div> <div><font face="Arial" size="2">It is my understanding the ICOOL/ECALC presently uses the first of these transformations, I.e., it uses coordinates</font></div> <div><font face="Arial" size="2"> <div><font face="Arial" size="2">(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)</font></div> </font></div> </div> </div> </div> </div> </div> </div> </div> <div> </div> <div><font face="Arial" size="2">[Rick Fernow: Can you comment on this?]</font></div> <div> </div> <div><font face="Arial" size="2">It now seems to me that this is actually OK in principle (although perhaps a somewhat "ugly" transformation)</font></div> <div> </div> <div><font face="Arial" size="2">3.   If we have potentials due to several different charge/current sources, I believe that it is OK to use different gauges for the different sources, if this proves to be computationally convenient.</font></div> <div> </div> <div><font face="Arial" size="2">It is less clear to me that we have the freedom to include the potentials from some sources, but to leave then out for others.</font></div> <div> </div> <div><font face="Arial" size="2">4.  There is a special gauge called the Hamiltonian gauge in which the scalar potential is everywhere zero.</font></div> <div> </div> <div><font face="Arial" size="2">See sec. 8 of </font></div> <div><font face="Arial" size="2"><a moz-do-not-send="true" title="http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf CTRL + Click to follow link" href="http://puhep1.princeton.edu/%7Emcdonald/examples/EM/jackson_ajp_70_917_02.pdf">http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf</a></font></div> <div> </div> <div><font face="Arial" size="2">user: archive</font></div> <div><font face="Arial" size="2">pass alpha137</font></div> <div> </div> <div><font face="Arial" size="2">However, for time-dependent fields it is not so easy to calculate the vector potential in this gauge.   So, I doubt that we will use it much.</font></div> <div> </div> <div><font face="Arial" size="2">[For static electric fields, V = 0 and A = - E t.    For static magnetic fields the vector potential is the same as the Coulomb gauge potential (which is the same as the Lorenz gauge potential for this case).]</font></div> <div> </div> <div><font face="Arial" size="2">5.  If we include the potentials in our emittance calculations, we will have to learn to live with the fact that the potentials in most gauges can be nonzero in regions where the fields E and B are zero.   In these regions, where the particle motion is unaffected by E and B, the emittance calculation will nonetheless depend on the potentials.   In principle, this should make no difference -- as we should get the same value for the emittance even if we neglect the potentials altogether.</font></div> <div> </div> <div><font face="Arial" size="2">6.  Juan Gallardo points out that if the E and B fields are known, the simplest calculation of corresponding potentials is in the so-called Poincare gauge.</font></div> <div> </div> <div><font face="Arial" size="2">See sec. 9 of Jackson's paper linked above.</font></div> <div> </div> <div><font face="Arial" size="2">In this calculation, the potentials depend on the E and B fields only along a line from the origin to the point in question.</font></div> <div> </div> <div><font face="Arial" size="2">This means that the value of the potentials depends on the choice of the origin.</font></div> <div> </div> <div><font face="Arial" size="2">If the origin is in a region of nonzero E or B field, we immediately see that the potentials in the Poincare gauge are nonzero throughout all space.</font></div> <div> </div> <div><font face="Arial" size="2">Juan and I have been making a few toy analytic calculations of potentials in the Poincare gauge to get some experience as to how it works:</font></div> <div><font face="Arial" size="2"><a moz-do-not-send="true" title="http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf CTRL + Click to follow link" href="http://puhep1.princeton.edu/%7Emcdonald/examples/cavity.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf</a></font></div> <div><font face="Arial" size="2"><a moz-do-not-send="true" title="http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf CTRL + Click to follow link" href="http://puhep1.princeton.edu/%7Emcdonald/examples/solpot.pdf">http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf</a></font></div> <div><font face="Arial" size="2"><a moz-do-not-send="true" title="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf CTRL + Click to follow link" href="http://puhep1.princeton.edu/%7Emcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</a></font></div> <div><font face="Arial" size="2"><a moz-do-not-send="true" title="http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf CTRL + Click to follow link" href="http://puhep1.princeton.edu/%7Emcdonald/examples/axial.pdf">http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf</a></font></div> <div><font face="Arial" size="2">[This is a work in process.  These notes are still being updated.</font></div> <div> </div> <div><font face="Arial" size="2">A price of using the Poincare gauge to simplify the calculation of the potentials will be that they must be calculated over the entire volume where our beams go.</font></div> <div> </div> <div><font face="Arial" size="2">The emittance calculation in the final cooling section must include the vector potential from the magnets back at the target station, etc.</font></div> <div> </div> <div><font face="Arial" size="2">I believe this complexity can be mitigated by choosing different origins to calculate the potentials from different field regions.</font></div> <div> </div> <div><font face="Arial" size="2">7.  All this discussion emphasizes how it would be simpler if we just omit the potentials when calculating the emittance.</font></div> <div> </div> <div><font face="Arial" size="2">Swann's "theorem" shows that this is OK for phase volume, but it may not be OK for calculations of rms emittance.</font></div> <div> </div> <div><font face="Arial" size="2">Challenge:  Give versions of Swann's theorem for rms emittance calculations.  Clarify whether the theorem holds only for 6D emittance, or for 4D and 2D subemittances as well.</font></div> <div> </div> <div><font face="Arial" size="2">Alexander Shemyakin claims that this has been done for the 4D transverse rms emittance for beams inside an ideal DC solenoids, but perhaps not for general electromagnetic fields.   Apparently, Alexey Burov does not believe the theorem holds even for this special case.</font></div> <div> </div> <div><font face="Arial" size="2">However, I get the impression from comments by Alex Dragt that rms emittances, are not, in general, invariants with respect to the independent variable of the Hamiltonian, be that either t or z.   So it may well be that rms emittances are not invariant under gauge transformations, or under transformations from canonical to noncanonical variables (I.e., neglecting the potentials).</font></div> <div> </div> <div><font face="Arial" size="2">Worry:  If rms emittances are not gauge invariant, then we have a whole new class of issues to deal with.</font></div> <div> </div> <div><font face="Arial" size="2">--Kirk </font></div> <div> </div> <div style="font: 10pt Tahoma;"> <div><br> </div> <div style="background: none repeat scroll 0% 0% rgb(245, 245, 245);"> <div style=""><b>From:</b> <a moz-do-not-send="true" title="mailto:dragtnb@comcast.net CTRL + Click to follow link" href="mailto:dragtnb@comcast.net">alex dragt</a> </div> <div><b>Sent:</b> Saturday, March 12, 2011 11:52 AM</div> <div><b>To:</b> <a moz-do-not-send="true" title="mailto:kirkmcd@Princeton.EDU CTRL + Click to follow link" href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</a> </div> <div><b>Cc:</b> <a moz-do-not-send="true" title="mailto:map-l@lists.bnl.gov CTRL + Click to follow link" href="mailto:map-l@lists.bnl.gov">MAP List</a> ; <a moz-do-not-send="true" title="mailto:rdryne@lbl.gov CTRL + Click to follow link" href="mailto:rdryne@lbl.gov">Robert D Ryne</a> ; <a moz-do-not-send="true" title="mailto:dragtnb@comcast.net CTRL + Click to follow link" href="mailto:dragtnb@comcast.net">alex dragt</a> </div> <div><b>Subject:</b> Re: [MAP] Use of x,y,t as Hamiltonian coordinates.</div> </div> </div> <div><br> </div> <font class="Apple-style-span" size="5"><span style="font-size: 18px;" class="Apple-style-span">Dear Kirk,  </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">     If you employ (1.5.29) and (1.5.30) in my book, you will find that</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">H=\gamma mc^2+q\psi.</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">Then. from (1.6.5), it follows that</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">p_t=-H=-\gamma mc^2-q\psi,</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">in agreement with Rob's slide 16.</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">Best,</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">Alex<br> </span></font> <div> </div> <div>If you </div> <div><br> <div> <div>On Mar 11, 2011, at 8:10 PM, Kirk T McDonald wrote:</div> <br class="Apple-interchange-newline"> <blockquote type="cite"> <div style="padding-left: 10px; padding-right: 10px; word-wrap: break-word; padding-top: 15px;" id="MailContainerBody" name="Compose message area" leftmargin="0" topmargin="0" bgcolor="#ffffff" canvastabstop="true"> <div><font face="Arial" size="2">Rob,</font></div> <div> </div> <div><font face="Arial" size="2">Thanks for this comment.</font></div> <div> </div> <div><font face="Arial" size="2">Your slide 16 seems to imply that the canonical momentum associated with coordinate t, when using (x,y,t) as coordinates, is</font></div> <div><font face="Arial" size="2">p_t = - (E_mech + q V).</font></div> <div> </div> <div><font face="Arial" size="2">This does not quite match what I infer from Alex Dragt that</font></div> <div> </div> <div><font face="Arial" size="2">p_t = - H</font></div> <div><font face="Arial" size="2">= - { sqrt[ m^2 c^4 + (p_mech - q A / c)^2 ] + q V }</font></div> <div> </div> <div><font face="Arial" size="2">How did you arrive at your simplification?</font></div> <div> </div> <div><font face="Arial" size="2">Your result matches Dragt's if the vector potential is zero.....</font></div> <div> </div> <div><font face="Arial" size="2">Are you saying that we can ignore the vector potential, but not the scalar potential?</font></div> <div> </div> <div><font face="Arial" size="2">--Kirk</font></div> <div style="font: 10pt Tahoma;"> <div><br> </div> <div style="background: none repeat scroll 0% 0% rgb(245, 245, 245);"> <div style=""><b>From:</b> <a moz-do-not-send="true" title="mailto:rdryne@lbl.gov CTRL + Click to follow link" href="mailto:rdryne@lbl.gov">Robert D Ryne</a> </div> <div><b>Sent:</b> Saturday, March 12, 2011 12:07 AM</div> <div><b>To:</b> <a moz-do-not-send="true" title="mailto:kirkmcd@Princeton.EDU CTRL + Click to follow link" href="mailto:kirkmcd@Princeton.EDU">Kirk T McDonald</a> </div> <div><b>Cc:</b> <a moz-do-not-send="true" title="mailto:dragtnb@comcast.net CTRL + Click to follow link" href="mailto:dragtnb@comcast.net">alex dragt</a> ; <a moz-do-not-send="true" title="mailto:map-l@lists.bnl.gov CTRL + Click to follow link" href="mailto:map-l@lists.bnl.gov">MAP List</a> </div> <div><b>Subject:</b> Re: [MAP] Use of x,y,t as Hamiltonian coordinates.</div> </div> </div> <div><br> </div> <span style="font-size: 14px;" class="Apple-style-span">Kirk,</span> <div style="font-size: 14px;"><br> </div> <div style="font-size: 14px;">The 6-vector of canonical variables is shown on slide 16 of my presentation at the MAP meeting. These are the variables that should be used for eigen-emittance calculations. Of course this happens "for free" in a code that uses canonical variables. When I calculate eigen-emittances from a non-canonical code, the diagnostic subroutine does the conversion to canonical variables.</div> <div style="font-size: 14px;"><br> </div> <div style="font-size: 14px;">Rob</div> <div style="font-size: 16px;"><span style="font-size: medium;" class="Apple-style-span"><font class="Apple-style-span" size="4"><span style="font-size: 16px;" class="Apple-style-span"><br> </span></font></span></div> <div><br> </div> </div> </blockquote> </div> <br> </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> </body> </html>