<HTML><HEAD> <META content="text/html; charset=ISO-8859-1" http-equiv=Content-Type></HEAD> <BODY dir=ltr bgColor=#66ffff text=#000000> <DIV dir=ltr> <DIV style="FONT-FAMILY: 'Arial'; COLOR: #000000; FONT-SIZE: 10pt"> <DIV>Folks</DIV> <DIV> </DIV> <DIV>I believe that what Alex Dragt has been trying to tell us is that for applications involving Hamiltonian dynamics it is most favored to use the so-called Hamiltonian gauge, in which the scalar potential V is zero everywhere.   Further, this ties in nicely with the use of coordinates (x,y,t) rather than (x,y,z).</DIV> <DIV> </DIV> <DIV>For a general expression for the vector potential A in this gauge, see sec. 8 of</DIV> <DIV><A title=http://puhep1.princeton.edu/~mcdonald/examples/EM/jackson_ajp_70_917_02.pdf 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></DIV> <DIV> </DIV> <DIV>For fields with time dependence e^{- i omega t), with nonzero wave number k = omega / c, then the vector potential is simply</DIV> <DIV>A = – i E / k.   (Gaussian units).</DIV> <DIV>Note that A = 0 wherever E = 0, in which regions B can only be static.</DIV> <DIV> </DIV> <DIV>For a static magnetic field, the vector potential is the same as that in the Coulomb gauge (and in the Lorenz gauge).  [In some static cases, such a toroidal magnets, the static vector potential will be nonzero in regions where B is zero; but in regions where a Fourier component B_omega is zero for nonzero omega, A_omega will be zero also.]</DIV> <DIV> </DIV> <DIV>A static electric field has the vector potential </DIV> <DIV>A = – c(t – t_0) E.</DIV> <DIV>This seems a bit odd, but will not bother us.</DIV> <DIV> </DIV> <DIV>An unusual feature of the Hamiltonian gauge vector potential is that is is not continuous at a perfectly conducting surface – which did bother me for quite a while.</DIV> <DIV> </DIV> <DIV>However, in using A to compute B, the (ill-defined) normal derivative at a perfectly conducting surface is not needed to deduce the tangential B.   </DIV> <DIV> </DIV> <DIV>Of course, the Hamiltonian gauge vector potential makes the canonical momentum p = p_mech + q A / c</DIV> <DIV>discontinuous at the surface of the rf cavity.</DIV> <DIV> </DIV> <DIV>This is troublesome if we use coordinates (x,y,z) and have an accelerating cavity with E_z, such that p_z takes a step on entering or leaving the cavity.</DIV> <DIV> </DIV> <DIV>However, if we switch to coordinates (x,y,t) the t-canonical momentum is</DIV> <DIV>p_t = – E_mech – q V = – E_mech</DIV> <DIV>in the Hamiltonian gauge, which is continuous at the cavity wall.</DIV> <DIV>Also, canonical momenta p_x and p_y are continuous if the particles enter and leave the cavity through faces at constant z (of a good conducting material).</DIV> <DIV> </DIV> <DIV>All this is serendipitous for the cooling sections of a muon collider, where we are almost certain to use cavities with flat (Be) faces where the particles enter and exit.</DIV> <DIV> </DIV> <DIV>It remains that rms emittance (and eigen-emittance, I believe) is not gauge invariant.   But we have to choose some gauge to proceed.</DIV> <DIV> </DIV> <DIV>It now looks like the Hamiltonian gauge is the one to use, along with coordinates (x,y,t) rather than (x,y,z).</DIV> <DIV> </DIV> <DIV>1. Time-dependent A_omega follows immediately from knowledge of E_omega, and is zero where E_omega and B_omega are zero.</DIV> <DIV> </DIV> <DIV>2.  Static A is same as in the Coulomb gauge (which is what we almost always use now).</DIV> <DIV> </DIV> <DIV>3.  No steps in canonical momenta so long as all rf cavities have flat faces where particles enter/exit.</DIV> <DIV> </DIV> <DIV>4.  No scalar potential to worry about, so p_t = – E_mech, as in our present software. (It doesn’t hurt anything to define p_t = + E_mech, as we actually do.)</DIV> <DIV> </DIV> <DIV>---------------------------------------------------------</DIV> <DIV>5.  Although phase volume is the same where or not we include the potentials in the momenta, this is not true for rms emittance (or eigen-emittance).   So we should stop using ECAL9, and switch to emitcalc (with a better approximation to the vector potential) and/or equivalent programs developed by Tom Roberts, Chris Rogers, et al.</DIV> <DIV> </DIV> <DIV>--Kirk</DIV></DIV></DIV></BODY></HTML>