<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN"> <HTML><HEAD> <META content=text/html;charset=iso-8859-1 http-equiv=Content-Type> <META name=GENERATOR content="MSHTML 8.00.6001.19019"></HEAD> <BODY style="PADDING-LEFT: 10px; PADDING-RIGHT: 10px; WORD-WRAP: break-word; PADDING-TOP: 15px; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space" id=MailContainerBody leftMargin=0 topMargin=0 bgColor=#ffffff CanvasTabStop="true" name="Compose message area"> <DIV><FONT size=2 face=Arial>Alex,</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>Thanks for that useful clarification.  I now agree that Rob's slide 16 is correct.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>This leads to several other comments.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>If you find this tedious, but are willing to consider some of the argument, skip to item 7.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>1.  With H = E_mech + q V,</FONT></DIV> <DIV><FONT size=2 face=Arial>the partial derivative on the diagonal of the Jacobian of the transformation from </FONT></DIV> <DIV><FONT size=2 face=Arial>canonical coordinates</FONT></DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x+qA_x/c, p_mech_y_qA_y/c, H=E_mech+qV)</FONT></DIV> <DIV><FONT size=2 face=Arial>to (noncanonical) coordinates</FONT></DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech,x, p_mech,y, E_mech)</FONT></DIV> <DIV><FONT size=2 face=Arial>is</FONT></DIV> <DIV><FONT size=2 face=Arial>partial H / partial E_mech = 1.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>2.  If we choose to include the potentials A and V in the calculation, we must do so consistently.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>However, we can give a corollary to Swann's theorem which indicates that in principle we have a lot of flexibility.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>For example, if we decide to change from the canonical coordinates</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, H=E_mech+qV)</FONT></DIV> <DIV><FONT size=2 face=Arial>to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x+qA_x/c, p_mech_y+qA_y/c, E_mech)</FONT></DIV> <DIV><FONT size=2 face=Arial>or to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x+qA_x/c, p_mech_y, H=E_mech+qV)</FONT></DIV> <DIV><FONT size=2 face=Arial>or to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x, p_mech_y+qA_y/c, H=E_mech+qV)</FONT></DIV> <DIV><FONT size=2 face=Arial>or to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x+qA_x/c, p_mech_y, E_mech)</FONT></DIV> <DIV><FONT size=2 face=Arial>or to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x, p_mech_y+qA_y/c, E_mech)</FONT></DIV> <DIV><FONT size=2 face=Arial>or to</FONT></DIV> <DIV> <DIV><FONT size=2 face=Arial>(x, y, t, p_mech_x, p_mech_y, H=E_mech+qV),</FONT></DIV> <DIV><FONT size=2 face=Arial>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 size=2 face=Arial>(x, y, t, p_mech,x, p_mech,y, E_mech) ].</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>[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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>It is my understanding the ICOOL/ECALC presently uses the first of these transformations, I.e., it uses coordinates</FONT></DIV> <DIV><FONT size=2 face=Arial> <DIV><FONT size=2 face=Arial>(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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>[Rick Fernow: Can you comment on this?]</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>It now seems to me that this is actually OK in principle (although perhaps a somewhat "ugly" transformation)</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>4.  There is a special gauge called the Hamiltonian gauge in which the scalar potential is everywhere zero.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>See sec. 8 of </FONT></DIV> <DIV><FONT size=2 face=Arial><A title="http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf
CTRL + Click to follow link" href="http://puhep1.princeton.edu/~mcdonald/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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>user: archive</FONT></DIV> <DIV><FONT size=2 face=Arial>pass alpha137</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>[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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>See sec. 9 of Jackson's paper linked above.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>This means that the value of the potentials depends on the choice of the origin.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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 size=2 face=Arial><A title="http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf
CTRL + Click to follow link" href="http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cavity.pdf</A></FONT></DIV> <DIV><FONT size=2 face=Arial><A title="http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf
CTRL + Click to follow link" href="http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf">http://puhep1.princeton.edu/~mcdonald/examples/solpot.pdf</A></FONT></DIV> <DIV><FONT size=2 face=Arial><A title="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf
CTRL + Click to follow link" href="http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf">http://puhep1.princeton.edu/~mcdonald/examples/cylindrical.pdf</A></FONT></DIV> <DIV><FONT size=2 face=Arial><A title="http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf
CTRL + Click to follow link" href="http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf">http://puhep1.princeton.edu/~mcdonald/examples/axial.pdf</A></FONT></DIV> <DIV><FONT size=2 face=Arial>[This is a work in process.  These notes are still being updated.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>I believe this complexity can be mitigated by choosing different origins to calculate the potentials from different field regions.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>7.  All this discussion emphasizes how it would be simpler if we just omit the potentials when calculating the emittance.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>Worry:  If rms emittances are not gauge invariant, then we have a whole new class of issues to deal with.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>--Kirk </FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV style="FONT: 10pt Tahoma"> <DIV><BR></DIV> <DIV style="BACKGROUND: #f5f5f5"> <DIV style="font-color: black"><B>From:</B> <A 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 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 title="mailto:map-l@lists.bnl.gov
CTRL + Click to follow link" href="mailto:map-l@lists.bnl.gov">MAP List</A> ; <A title="mailto:rdryne@lbl.gov
CTRL + Click to follow link" href="mailto:rdryne@lbl.gov">Robert D Ryne</A> ; <A 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; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space" id=MailContainerBody name="Compose message area" leftmargin="0" topmargin="0" bgcolor="#ffffff" canvastabstop="true"> <DIV><FONT size=2 face=Arial>Rob,</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>Thanks for this comment.</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>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 size=2 face=Arial>p_t = - (E_mech + q V).</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>This does not quite match what I infer from Alex Dragt that</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>p_t = - H</FONT></DIV> <DIV><FONT size=2 face=Arial>= - { sqrt[ m^2 c^4 + (p_mech - q A / c)^2 ] + q V }</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>How did you arrive at your simplification?</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>Your result matches Dragt's if the vector potential is zero.....</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>Are you saying that we can ignore the vector potential, but not the scalar potential?</FONT></DIV> <DIV><FONT size=2 face=Arial></FONT> </DIV> <DIV><FONT size=2 face=Arial>--Kirk</FONT></DIV> <DIV style="FONT: 10pt Tahoma"> <DIV><BR></DIV> <DIV style="BACKGROUND: #f5f5f5"> <DIV style="font-color: black"><B>From:</B> <A 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 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 title="mailto:dragtnb@comcast.net
CTRL + Click to follow link" href="mailto:dragtnb@comcast.net">alex dragt</A> ; <A 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></BODY></HTML>