Mikey
Nov18-04, 12:52 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nA question about the Hamilton-Jacobi theory has been bugging me for a while,\nhopefully someone here can put me out of my misery. If I understand\ncorrectly, the purpose of solving the time-independent HJ equation is that\nit allows us to convert to new canonical coordinates in which the\nHamiltonian only depends on the new momenta (say). Correct? So then the new\ncoordinates are all cyclic, and the new momenta are conserved. Not only\nthat, but the momenta all Poisson commute with each other, and with the\nHamiltonian (which is just a function of the new momenta). So it looks to me\nlike the system must be completely integrable. Does this mean you can only\nsolve the time-independent HJ eqn for completely integrable systems?\n\nI think I might be missing something here, so if someone could put me\nstraight that would be great. I have a feeling \'complete integrability\' has\nsomething to do with \'complete separability\' of the HJ eqn, but I\'m\nconfused.\n\nThanks for your help,\nMikey\n\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>A question about the Hamilton-Jacobi theory has been bugging me for a while,
hopefully someone here can put me out of my misery. If I understand
correctly, the purpose of solving the time-independent HJ equation is that
it allows us to convert to new canonical coordinates in which the
Hamiltonian only depends on the new momenta (say). Correct? So then the new
coordinates are all cyclic, and the new momenta are conserved. Not only
that, but the momenta all Poisson commute with each other, and with the
Hamiltonian (which is just a function of the new momenta). So it looks to me
like the system must be completely integrable. Does this mean you can only
solve the time-independent HJ eqn for completely integrable systems?
I think I might be missing something here, so if someone could put me
straight that would be great. I have a feeling 'complete integrability' has
something to do with 'complete separability' of the HJ eqn, but I'm
confused.
Thanks for your help,
Mikey
hopefully someone here can put me out of my misery. If I understand
correctly, the purpose of solving the time-independent HJ equation is that
it allows us to convert to new canonical coordinates in which the
Hamiltonian only depends on the new momenta (say). Correct? So then the new
coordinates are all cyclic, and the new momenta are conserved. Not only
that, but the momenta all Poisson commute with each other, and with the
Hamiltonian (which is just a function of the new momenta). So it looks to me
like the system must be completely integrable. Does this mean you can only
solve the time-independent HJ eqn for completely integrable systems?
I think I might be missing something here, so if someone could put me
straight that would be great. I have a feeling 'complete integrability' has
something to do with 'complete separability' of the HJ eqn, but I'm
confused.
Thanks for your help,
Mikey