View Full Version : Re: How to Field Theorize Starting from a Lagrangian
Hendrik van Hees
Apr7-04, 08:34 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\n\n\nFrank Hellmann wrote:\n\n> I think a ressource that would bring together several simple examples\n> of how variational methods work, some simple Lagrangians for different\n> (toy) applications and which would show how simple variational\n> calculations really are when approached from the right angel would be\n> much appreciated.\n\nI can only offer my (admittedly very short) appendix on the subject in\nmy qft script on my homepage. I hope this helps a little:\n\nhttp://theory.gsi.de/~vanhees/publ/lect.pdf\n\n--\nHendrik van Hees Cyclotron Institute\nPhone: +1 979/845-1411 Texas A&M University\nFax: +1 979/845-1899 Cyclotron Institute, MS-3366\nhttp://theory.gsi.de/~vanhees/ College Station, TX 77843-3366\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>Frank Hellmann wrote:
> I think a ressource that would bring together several simple examples
> of how variational methods work, some simple Lagrangians for different
> (toy) applications and which would show how simple variational
> calculations really are when approached from the right angel would be
> much appreciated.
I can only offer my (admittedly very short) appendix on the subject in
my qft script on my homepage. I hope this helps a little:
http://theory.gsi.de/~vanhees/publ/lect.pdf
--
Hendrik van Hees Cyclotron Institute
Phone: +1 979/845-1411 Texas A&M University
Fax: +1 979/845-1899 Cyclotron Institute, MS-3366
http://theory.gsi.de/~vanhees/ College Station, TX 77843-3366
tessel@um.bot
Apr7-04, 08:34 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>On Sun, 4 Apr 2004, Aaron Bergman wrote:\n\n> All the issues you raise are pretty scattered.\n\nWell, do you at least agree there\'s some point to gathering the relevant\nideas together in one place?\n\n> Goldstein (even though I didn\'t use it) almost assuredly covers\n> Lagrangians quite extensively.\n\nFine, but how many of the specific "issues for discussion" which I\nmentioned does this book cover? Also, I think I probably know which book\nyou have in mind, but can you please give a citation?\n\n(Sorry, I still haven\'t gotten around to reskimming Peskin and Shroeder,\nbut I promise to do this.)\n\n> It\'s not all done in one place in the curriculum always, but I haven\'t\n> really noticed a gap.\n\nMany thanks to Frank Hellman, who commented that he agrees that there -is-\nsomething of a gap, and gave another example of the kind of [serious]\n"Lagranian play" I had in mind. I also eagerly await any comments by\nSteve Carlip, e.g. regarding searching for an EIH analog.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Sun, 4 Apr 2004, Aaron Bergman wrote:
> All the issues you raise are pretty scattered.
Well, do you at least agree there's some point to gathering the relevant
ideas together in one place?
> Goldstein (even though I didn't use it) almost assuredly covers
> Lagrangians quite extensively.
Fine, but how many of the specific "issues for discussion" which I
mentioned does this book cover? Also, I think I probably know which book
you have in mind, but can you please give a citation?
(Sorry, I still haven't gotten around to reskimming Peskin and Shroeder,
but I promise to do this.)
> It's not all done in one place in the curriculum always, but I haven't
> really noticed a gap.
Many thanks to Frank Hellman, who commented that he agrees that there -is-
something of a gap, and gave another example of the kind of [serious]
"Lagranian play" I had in mind. I also eagerly await any comments by
Steve Carlip, e.g. regarding searching for an EIH analog.
"T. Essel" (hiding somewhere in cyberspace)
tessel@um.bot
Apr7-04, 08:35 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>(Attention please Steve Carlip!!! Do you agree with Aaron and Shagird\nthat the issues I raised are adequately covered in an existing textbook?)\n\nOn Sun, 4 Apr 2004, Buzurg Shagird wrote:\n\n> tessel@tum.bot wrote in message news:<c4kqmp$18q3$1@fiasco.xenopsyche.net>...\n>\n> > I take it you feel there is in fact no such "gap" as I imagined? Do\n> > other physics faculty here agree with that? If so, I\'ll gladly cease\n> > and desist.\n>\n> I am not sure where you see that gap ... surely Landau and Lifshitz do a\n> fairly good job of discussing the action principle in their Classical\n> Mechanics and Classical Field Theory books?\n\nSince I mentioned those books as inadequate for students like Doug\nSweetser who are trying to formulate and play around with a "Lagrangian\nfield theory" (even a "toy theory"), you may assume that I feel they (and\nthe other books I mentioned) do -not- explain -the "issues for discussion"\nwhich I listed- (see below for a repeat of this list).\n\n> Most quantum field theory textbooks also do a quick review of the\n> classical field theory results they are going to use, like the\n> Euler-Lagrange equations, Noether\'s theorem, Hamiltonian and conserved\n> quantities ...\n\nLet me clarify: it is true that LL and other books I have studied -do-\ndiscuss the following topics:\n\n(1) the E-L equations for L containing only first derivatives,\n\n(2) a special case of Noether\'s theorem (physicists may not know the more\ngeneral statement using variational symmetries or even "divergence\nsymmetries", for which see Olver),\n\n(3) Hamiltonian and conservation of energy,\n\n(4) EM tensor for a very few and very special cases (e.g. scalar fields,\nMaxwell, linearized gtr) and conservation of (linear, angular) momentum.\n\nSince I\'ve studied those books, I know they discuss these topics and I did\nnot mean to imply that I feel that their discussion of -these- topics is\ninadequate!\n\nMy claim is quite different: for purposes of playing around intelligently\nwith Lagrangians, -much more is needed-.\n\nPerhaps most obviously, as I pointed out, if you try to perturb your\nfavorite field theory Lagrangian, you will probably encounter a\n-nonlinear- field equation. For example, the sine-Gordon equation can\nappear in this way. Now, since an early step in investigating a proposed\nfield theory is to search for "wave solutions", you will need tools for\ndoing this even in the case of a -nonlinear- PDE. Average students who\nhave only studied -linear- PDEs (e.g. solution by separation of variables)\nare likely to have great difficulty in doing that, but Lie\'s methods are\nstraightforward and easy to use (the intermediate computations can become\ntedious, but heck, that\'s why people invented me).\n\nOr again: when I proposed to try to derive the equation of motion of\n"charged" test particles in a given field, did you and Aaron fail to\nnotice that I was talking about the analogue of the EIH procedure in the\ncase of gtr? (For other readers: Einstein, Infeld, and Hoffmann proposed\nto remove the assumption that the world lines nonspinning test particles\nin a vacuum region are timelike geodesics by -deriving- this from the\nEinstein field equation.) I assume that you and Aaron are both aware that\nthis is -highly nontrivial-; if not, see\n\nauthor = {Joshua N. Goldberg},\ntitle = {The Equations of Motion},\nbooktitle = {Gravitation: an Introduction to Current Research},\neditor = {Louis Witten},\npublisher = {Wiley},\npages = {102--129},\nyear = 1962}\n\nNote that I was proposing to explore whether an analogous procedure (for\nobtaining the equation of motion of "charged" particles from the field\nequation) can be found for other theories. Another way to think about it:\ncan we determine a suitable (particle-field) interaction term from a\nproposal for a free-field term in the Lagrangian of our field theory? This\nis presumably a nontrivial question!\n\n> You could complain that Dirac\'s method of analysis of constraints is not\n> given in more books,\n\nI didn\'t mention that, probably because I haven\'t heard of it, at least\nnot under that name.\n\n> but you can\'t apply it directly to non-Abelian gauge theories anyway ---\n> you need to go to superspace or such to introduce ghosts, and ghosts are\n> better done in the path integral formalism anyway -- so you shouldn\'t\n> really complain. :-)\n\nI didn\'t complain about those issues. Indeed, I forgot to say that one of\nthe "levels of structure" I had in mind was:\n\n(5) first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge\nsymmetries.\n\n> What else will fill this \'gap\' that you are worrying about?\n\nHonestly, at this point I have to wonder if you, Aaron even -read- my post\nin its entirety! In particular, did either of you read the list of\n"issues for discussion" that I offered? Did either of you review the\ncontents of the books I mentioned with this list in mind? Did either of\nyou see any of Doug\'s efforts, or the efforts of other amateurs here over\nthe years, to create some kind of "toy theory" starting from a proposed\nLagrangian? I am an amateur myself, but even I could spot zillions of\nerrors committed by these people, some quite elementary and some involving\nmore subtle issues. For that matter, look at the ArXiv--- I have seen\nquite a few poor quality papers there committing similar errors (but don\'t\nask me for examples, because I\'ve deleted my "file of horrors" from the\nArXiv, not to mention that could get ugly--- my interest is not in cutting\ndown people but in preventing others from committing similar fundamental\nerrors, in the ultimate interest of promoting the appearance of better and\nthus more interesting papers).\n\nThus my suspicion that average students, and perhaps even some folk\nsubmitting papers to the ArXiv, need more help than offered by\nLandau/Lifshitz or Ohanian/Ruffini.\n\nFor the readers convenience, here again is my list of proposed "issues for\ndiscussion":\n\n============== BEGIN SELF-QUOTATION ====================\n\nI propose to start as simply as possible and slowly increase the level of\nsophistication, e.g. (not neccessarily in this order)\n\n1. first linear, then nonlinear Lagrangians,\n\n2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more\nvariables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),\n\n3. first scalar theories, then vector theories, then tensor theories,\n\n4. first classical field theories, then quantum field theories.\n\n[5. first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge\nsymmetries.]\n\nIssues we might systematically explore in this way could include:\n\n1. how to sensibly select the terms of our Lagrangian, as written in the\nuseful form\n\nL = L_matt + L_interaction + L_field\n\n2. how to guess key features of the "free-field" case of the field\nequation of our theory from looking over L_field,\n\n3. how to write down the Euler-Lagrange equations (allowing for the\npossibility of second and higher order derivatives),\n\n4. how to verify key features of the (free-field) field equation, e.g.\n\n(a) how to determine the group of "point symmetries" and the subgroup of\n"variational symmetries",\n\n(b) how to confirm the existence of wavelike solutions to the (free-field)\nfield equation,\n\n(c) how to obtain and study particular solutions (even for nonlinear PDEs)\ne.g. using Lie\'s methods, Baecklund symmetries, scattering transform,\n\n5. how to obtain a suitable energy-momentum tensor (and what to do if the\n"canonical energy-momentum tensor" turns out to be -nonsymmetric-),\n\n6. more generally how to find "conserved currents" (e.g. from "divergence\nsymmetries"), at least in the case where L only contains first order\nderivatives,\n\n7. how to find suitable "conserved currents" if L contains second order\nderivatives,\n\n8. how to determine L_interaction from energy-momentum exchange between\nthe field and "particles", if this is possible, or if not, how otherwise\nto obtain a suitable "equation of motion" for "charged" particles,\n\n9. how to check for at least the most commonly encountered\nself-inconsistencies which can arise in a field theory,\n\nHaving explored these issues for a generous range of simple toy theories,\nwe can try to address the question which presumably we would all agree is\nthe question of ultimate interest: how can we write down a Lagrangian\nwhich has a reasonable chance of producing a theory which will accomplish\nstated goals?\n\nLet me mention a specific example of a "stated goal". Recall that\nMaxwell\'s theory of EM is beset with singularities in the field. Recall\nalso that a few years ago we had some people here who wanted to replace\ngtr with a theory which would "ameliorate" singularities while preserving\nthe remarkable experimental/observational success of that theory. In\nprinciple, as I said at the time, that\'s not a bad idea, but I claimed\nthen (and repeat now) that this is a -much- more challenging program than\n(I felt) those people recognized. At that time I proposed to step back and\nconsider first the analogous but easier question for EM, but unfortunately\n(as I recall the affair, at least) my correspondents were insufficiently\nwilling to do this. I hope and believe that the current student\ngeneration is more amenable to such good advice!\n\nActually, I propose to start with scalar theories, e.g.\nthree-dimensional analogues of\n\nL = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ]\n\n(I might have munged a sign or two, but this is supposed to be the\nLagrangian which leads to the sine-Gordon equation, a nonlinear wave\nequation beloved of PDE people), or more generally\n\nL = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term"\nwhich is small for small u\n\n(since I doubt the sine-Gordon equation will fully achieve the goal of\n"ameliorizating" singularities in our scalar field).\n\n============== END SELF-QUOTATION =======================\n\nI wonder if either Shagird or Aaron really think that average students\nwill be able to attack successfully the example of "stated goal" which I\nmentioned without help well beyond that offered by Landau/Lifshitz or\nOhanian/Rufinni? Or the question of searching for an analogue to the\nproposal of EIH? Or even searching for "wave solutions" to a nonlinear\nfree-field equation like the sine-Gordon equation?\n\nIf so, I challenge either of them to show us/them how to do this! Or,\nmuch more modestly, to cite chapter and verse from an -existing textbook-\nwhere this is done, in enough detail for an average student to follow the\ndiscussion.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>(Attention please Steve Carlip!!! Do you agree with Aaron and Shagird
that the issues I raised are adequately covered in an existing textbook?)
On Sun, 4 Apr 2004, Buzurg Shagird wrote:
> tessel@tum.bot wrote in message news:<c4kqmp$18q3$1@fiasco.xenopsyche.net>...
>
> > I take it you feel there is in fact no such "gap" as I imagined? Do
> > other physics faculty here agree with that? If so, I'll gladly cease
> > and desist.
>
> I am not sure where you see that gap ... surely Landau and Lifshitz do a
> fairly good job of discussing the action principle in their Classical
> Mechanics and Classical Field Theory books?
Since I mentioned those books as inadequate for students like Doug
Sweetser who are trying to formulate and play around with a "Lagrangian
field theory" (even a "toy theory"), you may assume that I feel they (and
the other books I mentioned) do -not- explain -the "issues for discussion"
which I listed- (see below for a repeat of this list).
> Most quantum field theory textbooks also do a quick review of the
> classical field theory results they are going to use, like the
> Euler-Lagrange equations, Noether's theorem, Hamiltonian and conserved
> quantities ...
Let me clarify: it is true that LL and other books I have studied -do-
discuss the following topics:
(1) the E-L equations for L containing only first derivatives,
(2) a special case of Noether's theorem (physicists may not know the more
general statement using variational symmetries or even "divergence
symmetries", for which see Olver),
(3) Hamiltonian and conservation of energy,
(4) EM tensor for a very few and very special cases (e.g. scalar fields,
Maxwell, linearized gtr) and conservation of (linear, angular) momentum.
Since I've studied those books, I know they discuss these topics and I did
not mean to imply that I feel that their discussion of -these- topics is
inadequate!
My claim is quite different: for purposes of playing around intelligently
with Lagrangians, -much more is needed-.
Perhaps most obviously, as I pointed out, if you try to perturb your
favorite field theory Lagrangian, you will probably encounter a
-nonlinear- field equation. For example, the sine-Gordon equation can
appear in this way. Now, since an early step in investigating a proposed
field theory is to search for "wave solutions", you will need tools for
doing this even in the case of a -nonlinear- PDE. Average students who
have only studied -linear- PDEs (e.g. solution by separation of variables)
are likely to have great difficulty in doing that, but Lie's methods are
straightforward and easy to use (the intermediate computations can become
tedious, but heck, that's why people invented me).
Or again: when I proposed to try to derive the equation of motion of
"charged" test particles in a given field, did you and Aaron fail to
notice that I was talking about the analogue of the EIH procedure in the
case of gtr? (For other readers: Einstein, Infeld, and Hoffmann proposed
to remove the assumption that the world lines nonspinning test particles
in a vacuum region are timelike geodesics by -deriving- this from the
Einstein field equation.) I assume that you and Aaron are both aware that
this is -highly nontrivial-; if not, see
author = {Joshua N. Goldberg},
title = {The Equations of Motion},
booktitle = {Gravitation: an Introduction to Current Research},
editor = {Louis Witten},
publisher = {Wiley},
pages = {102--129},
year = 1962}
Note that I was proposing to explore whether an analogous procedure (for
obtaining the equation of motion of "charged" particles from the field
equation) can be found for other theories. Another way to think about it:
can we determine a suitable (particle-field) interaction term from a
proposal for a free-field term in the Lagrangian of our field theory? This
is presumably a nontrivial question!
> You could complain that Dirac's method of analysis of constraints is not
> given in more books,
I didn't mention that, probably because I haven't heard of it, at least
not under that name.
> but you can't apply it directly to non-Abelian gauge theories anyway ---
> you need to go to superspace or such to introduce ghosts, and ghosts are
> better done in the path integral formalism anyway -- so you shouldn't
> really complain. :-)
I didn't complain about those issues. Indeed, I forgot to say that one of
the "levels of structure" I had in mind was:
(5) first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge
symmetries.
> What else will fill this 'gap' that you are worrying about?
Honestly, at this point I have to wonder if you, Aaron even -read- my post
in its entirety! In particular, did either of you read the list of
"issues for discussion" that I offered? Did either of you review the
contents of the books I mentioned with this list in mind? Did either of
you see any of Doug's efforts, or the efforts of other amateurs here over
the years, to create some kind of "toy theory" starting from a proposed
Lagrangian? I am an amateur myself, but even I could spot zillions of
errors committed by these people, some quite elementary and some involving
more subtle issues. For that matter, look at the ArXiv--- I have seen
quite a few poor quality papers there committing similar errors (but don't
ask me for examples, because I've deleted my "file of horrors" from the
ArXiv, not to mention that could get ugly--- my interest is not in cutting
down people but in preventing others from committing similar fundamental
errors, in the ultimate interest of promoting the appearance of better and
thus more interesting papers).
Thus my suspicion that average students, and perhaps even some folk
submitting papers to the ArXiv, need more help than offered by
Landau/Lifshitz or Ohanian/Ruffini.
For the readers convenience, here again is my list of proposed "issues for
discussion":
============== BEGIN SELF-QUOTATION ====================
I propose to start as simply as possible and slowly increase the level of
sophistication, e.g. (not neccessarily in this order)
1. first linear, then nonlinear Lagrangians,
2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more
variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),
3. first scalar theories, then vector theories, then tensor theories,
4. first classical field theories, then quantum field theories.
[5. first Lagrangians admitting U(1) gauge symmetry, then nonabelian gauge
symmetries.]
Issues we might systematically explore in this way could include:
1. how to sensibly select the terms of our Lagrangian, as written in the
useful form
L = L_{matt} + L_{interaction} + L_{field}
2. how to guess key features of the "free-field" case of the field
equation of our theory from looking over L_{field},
3. how to write down the Euler-Lagrange equations (allowing for the
possibility of second and higher order derivatives),
4. how to verify key features of the (free-field) field equation, e.g.
(a) how to determine the group of "point symmetries" and the subgroup of
"variational symmetries",
(b) how to confirm the existence of wavelike solutions to the (free-field)
field equation,
(c) how to obtain and study particular solutions (even for nonlinear PDEs)
e.g. using Lie's methods, Baecklund symmetries, scattering transform,
5. how to obtain a suitable energy-momentum tensor (and what to do if the
"canonical energy-momentum tensor" turns out to be -nonsymmetric-),
6. more generally how to find "conserved currents" (e.g. from "divergence
symmetries"), at least in the case where L only contains first order
derivatives,
7. how to find suitable "conserved currents" if L contains second order
derivatives,
8. how to determine L_{interaction} from energy-momentum exchange between
the field and "particles", if this is possible, or if not, how otherwise
to obtain a suitable "equation of motion" for "charged" particles,
9. how to check for at least the most commonly encountered
self-inconsistencies which can arise in a field theory,
Having explored these issues for a generous range of simple toy theories,
we can try to address the question which presumably we would all agree is
the question of ultimate interest: how can we write down a Lagrangian
which has a reasonable chance of producing a theory which will accomplish
stated goals?
Let me mention a specific example of a "stated goal". Recall that
Maxwell's theory of EM is beset with singularities in the field. Recall
also that a few years ago we had some people here who wanted to replace
gtr with a theory which would "ameliorate" singularities while preserving
the remarkable experimental/observational success of that theory. In
principle, as I said at the time, that's not a bad idea, but I claimed
then (and repeat now) that this is a -much- more challenging program than
(I felt) those people recognized. At that time I proposed to step back and
consider first the analogous but easier question for EM, but unfortunately
(as I recall the affair, at least) my correspondents were insufficiently
willing to do this. I hope and believe that the current student
generation is more amenable to such good advice!
Actually, I propose to start with scalar theories, e.g.
three-dimensional analogues of
L = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ]
(I might have munged a sign or two, but this is supposed to be the
Lagrangian which leads to the sine-Gordon equation, a nonlinear wave
equation beloved of PDE people), or more generally
L = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term"
which is small for small u
(since I doubt the sine-Gordon equation will fully achieve the goal of
"ameliorizating" singularities in our scalar field).
============== END SELF-QUOTATION =======================
I wonder if either Shagird or Aaron really think that average students
will be able to attack successfully the example of "stated goal" which I
mentioned without help well beyond that offered by Landau/Lifshitz or
Ohanian/Rufinni? Or the question of searching for an analogue to the
proposal of EIH? Or even searching for "wave solutions" to a nonlinear
free-field equation like the sine-Gordon equation?
If so, I challenge either of them to show us/them how to do this! Or,
much more modestly, to cite chapter and verse from an -existing textbook-
where this is done, in enough detail for an average student to follow the
discussion.
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\n\n\n\ntessel@um.bot wrote:\n\n> Issues we might systematically explore in this way could include:\n>\n> 1. how to sensibly select the terms of our Lagrangian, as written in the\n> useful form\n>\n> L = L_matt + L_interaction + L_field\n\n\nI think this format for the Lagrangian is found in gauge theories. this\nis a restricted class of theories, but an important one. For gauge\ntheories, it is easy to write the Lagrangian in the way you ask, you can\nfind it in any book on gauge theory.\n\nBut for more general theories, the Lagrangian won\'t take this form. or\nat least, there isn\'t a distinction between the "matter field" and the\n"field field". you can always seperate the linear terms from the\nnonlinear terms and call the former L_fields and the latter L_interaction.\n\n\n>\n> 2. how to guess key features of the "free-field" case of the field\n> equation of our theory from looking over L_field,\n\nWhich key features about the free-field version of your theory do you\nwant to guess? In general, i don\'t think free fields have all that many\nfeatures. It seems to me like you can know just about everything you\nmight want to know about a free field if you know its mass and spin.\n\n>\n> 3. how to write down the Euler-Lagrange equations (allowing for the\n> possibility of second and higher order derivatives),\n\nI recall this homework question in Goldstein. It is easy to write down\nthe Euler-Lagrange equation for Lagrangians with higher derivatives, and\nnot particularly enlightening.\n\nOne thing I wouldn\'t mind knowing more about: as i understand it, the\nreason we do not use Lagrangians with higher order derivatives is that\nthey would violate locality. This is, for example, the criterion by\nwhich we throw out the square root of the Klein-Gordon equation\n(d\\phi/dt=sqrt{d^2/dx^2+m^2}\\phi) as a useful theory in some field\ntheory textbooks. it contains derivatives of all orders, and so is\nnonlocal.\n\nif anyone wants to say why higher order derivatives violate locality, i\nwould love to hear it.\n\n>\n> 5. how to obtain a suitable energy-momentum tensor (and what to do if the\n> "canonical energy-momentum tensor" turns out to be -nonsymmetric-),\n\nFor a really nice treatment of the Belinfante procedure, and other\ndiscussions of the stress tensor, see Di Francesco, Conformal Field Theory.\n\n\n>\n> 6. more generally how to find "conserved currents" (e.g. from "divergence\n> symmetries"), at least in the case where L only contains first order\n> derivatives,\n\nI don\'t know what a divergence symmetry is. Does Noether\'s procedure\nnot work here?\n\n>\n> 7. how to find suitable "conserved currents" if L contains second order\n> derivatives,\n\nI don\'t see why we shouldn\'t be able to do Noether\'s procedure on a\nLagrangian with higher order derivatives....\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>tessel@um.bot wrote:
> Issues we might systematically explore in this way could include:
>
> 1. how to sensibly select the terms of our Lagrangian, as written in the
> useful form
>
> L = L_{matt} + L_{interaction} + L_{field}
I think this format for the Lagrangian is found in gauge theories. this
is a restricted class of theories, but an important one. For gauge
theories, it is easy to write the Lagrangian in the way you ask, you can
find it in any book on gauge theory.
But for more general theories, the Lagrangian won't take this form. or
at least, there isn't a distinction between the "matter field" and the
"field field". you can always seperate the linear terms from the
nonlinear terms and call the former L_{fields} and the latter L_{interaction}.
>
> 2. how to guess key features of the "free-field" case of the field
> equation of our theory from looking over L_{field},
Which key features about the free-field version of your theory do you
want to guess? In general, i don't think free fields have all that many
features. It seems to me like you can know just about everything you
might want to know about a free field if you know its mass and spin.
>
> 3. how to write down the Euler-Lagrange equations (allowing for the
> possibility of second and higher order derivatives),
I recall this homework question in Goldstein. It is easy to write down
the Euler-Lagrange equation for Lagrangians with higher derivatives, and
not particularly enlightening.
One thing I wouldn't mind knowing more about: as i understand it, the
reason we do not use Lagrangians with higher order derivatives is that
they would violate locality. This is, for example, the criterion by
which we throw out the square root of the Klein-Gordon equation
(d\phi/dt=\sqrt{d^2/dx^2+m^2}\phi) as a useful theory in some field
theory textbooks. it contains derivatives of all orders, and so is
nonlocal.
if anyone wants to say why higher order derivatives violate locality, i
would love to hear it.
>
> 5. how to obtain a suitable energy-momentum tensor (and what to do if the
> "canonical energy-momentum tensor" turns out to be -nonsymmetric-),
For a really nice treatment of the Belinfante procedure, and other
discussions of the stress tensor, see Di Francesco, Conformal Field Theory.
>
> 6. more generally how to find "conserved currents" (e.g. from "divergence
> symmetries"), at least in the case where L only contains first order
> derivatives,
I don't know what a divergence symmetry is. Does Noether's procedure
not work here?
>
> 7. how to find suitable "conserved currents" if L contains second order
> derivatives,
I don't see why we shouldn't be able to do Noether's procedure on a
Lagrangian with higher order derivatives....
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\n\n\n\nC.i.m@gmx.net (Frank Hellmann) wrote in message news:<e7f834be.0404041028.710dd415@posting.google.com>...\n>\n> I think a ressource that would bring together several simple examples\n> of how variational methods work, some simple Lagrangians for different\n> (toy) applications and which would show how simple variational\n> calculations really are when approached from the right angel would be\n> much appreciated.\n>\n> Cheers,\n> Frank.\n\nThere is a book called "Lagrangian Interaction", by Noel A. Doughty.\nI have it and it\'s a satisfying survey of the subject. I don\'t know\nhow much it would help a student, but you could take a look.\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>C.i.m@gmx.net (Frank Hellmann) wrote in message news:<e7f834be.0404041028.710dd415@posting.google.com>...
>
> I think a ressource that would bring together several simple examples
> of how variational methods work, some simple Lagrangians for different
> (toy) applications and which would show how simple variational
> calculations really are when approached from the right angel would be
> much appreciated.
>
> Cheers,
> Frank.
There is a book called "Lagrangian Interaction", by Noel A. Doughty.
I have it and it's a satisfying survey of the subject. I don't know
how much it would help a student, but you could take a look.
Danny Ross Lunsford
Apr7-04, 08:41 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\n\ntessel@um.bot wrote:\n\n[lament about lack of good references RE field theory]\n\nTwo book recommendations\n\n1) Gelfand-Fomin, "Calculus of Variations"\n\n2) Lovelock and Rund, "Tensors, Diff. Forms, and Variational Principles"\n\nThe latter has a large section at the end in which a combined\nvector-metric theory is examined in gory detail. The former is the rare\nmath book that really understands physics. We used it in my graduate\ncourse in the calculus of variations.\n\nI agree that there are a lot of papers on the arxiv that are poorly\nwritten noise. For this one can thank the komissar.\n\n-drl\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>tessel@um.bot wrote:
[lament about lack of good references RE field theory]
Two book recommendations
1) Gelfand-Fomin, "Calculus of Variations"
2) Lovelock and Rund, "Tensors, Diff. Forms, and Variational Principles"
The latter has a large section at the end in which a combined
vector-metric theory is examined in gory detail. The former is the rare
math book that really understands physics. We used it in my graduate
course in the calculus of variations.
I agree that there are a lot of papers on the arxiv that are poorly
written noise. For this one can thank the komissar.
-drl
tessel@um.bot
Apr7-04, 08:42 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\nI complained:\n\n> > [...] as far as I know, there is currently -no- textbook on "How to\n> > Theorize from a Lagrangian"\n\nOn Fri, 2 Apr 2004, MM replied:\n\n> I would have said that Peskin & Schroeder\'s "Introduction to QFT", and\n> Weinberg\'s multi-volume "The Quantum Theory of Fields" together do a\n> good job.\n\nOK, I\'ve made a mental note. But AFAIK neither of these two study the\nquestions I asked regarding -classical- field theories. Does anyone else\nagree that those questions are by no means trivial? I am beginning to\nwonder...\n\nI trust people who have the books I mentioned, namely\n\n1. Landau/Lifschitz, Mechanics,\n\n2. Landau/Lifschitz, Classical Theory of Fields),\n\n3. Ohanian/Ruffini, Gravitation and Spacetime,\n\n4. Weinberg, Gravitation and Cosmology\n\nwill agree that the questions I raised are largely unaddressed in these\nbooks, although from playing around I do understand why LL and OR make\ncertain comments which only apply in examples beyond the two examples\nthese books discuss.\n\nI expect (perhaps) naively that my questions should also be relevant in\nQFT, e.g. because I am vaguely aware of mathematical analogies between the\nusual suspects (PDEs arising in classical field theory and in quantum\ntheory). Is this expectation also wrong?\n\n"T. Essel" (hiding somewhere in cyberspace)\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>I complained:
> > [...] as far as I know, there is currently -no- textbook on "How to
> > Theorize from a Lagrangian"
On Fri, 2 Apr 2004, MM replied:
> I would have said that Peskin & Schroeder's "Introduction to QFT", and
> Weinberg's multi-volume "The Quantum Theory of Fields" together do a
> good job.
OK, I've made a mental note. But AFAIK neither of these two study the
questions I asked regarding -classical- field theories. Does anyone else
agree that those questions are by no means trivial? I am beginning to
wonder...
I trust people who have the books I mentioned, namely
1. Landau/Lifschitz, Mechanics,
2. Landau/Lifschitz, Classical Theory of Fields),
3. Ohanian/Ruffini, Gravitation and Spacetime,
4. Weinberg, Gravitation and Cosmology
will agree that the questions I raised are largely unaddressed in these
books, although from playing around I do understand why LL and OR make
certain comments which only apply in examples beyond the two examples
these books discuss.
I expect (perhaps) naively that my questions should also be relevant in
QFT, e.g. because I am vaguely aware of mathematical analogies between the
usual suspects (PDEs arising in classical field theory and in quantum
theory). Is this expectation also wrong?
"T. Essel" (hiding somewhere in cyberspace)
tessel@um.bot
Apr7-04, 08:42 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>\nI complained:\n\n> > [...] as far as I know, there is currently -no- textbook on "How to\n> > Theorize from a Lagrangian"\n\nOn Fri, 2 Apr 2004, MM replied:\n\n> I would have said that Peskin & Schroeder\'s "Introduction to QFT", and\n> Weinberg\'s multi-volume "The Quantum Theory of Fields" together do a\n> good job.\n\nOK, I\'ve made a mental note. But AFAIK neither of these two study the\nquestions I asked regarding -classical- field theories. Does anyone else\nagree that those questions are by no means trivial? I am beginning to\nwonder...\n\nI trust people who have the books I mentioned, namely\n\n1. Landau/Lifschitz, Mechanics,\n\n2. Landau/Lifschitz, Classical Theory of Fields),\n\n3. Ohanian/Ruffini, Gravitation and Spacetime,\n\n4. Weinberg, Gravitation and Cosmology\n\nwill agree that the questions I raised are largely unaddressed in these\nbooks, although from playing around I do understand why LL and OR make\ncertain comments which only apply in examples beyond the two examples\nthese books discuss.\n\nI expect (perhaps) naively that my questions should also be relevant in\nQFT, e.g. because I am vaguely aware of mathematical analogies between the\nusual suspects (PDEs arising in classical field theory and in quantum\ntheory). Is this expectation also wrong?\n\n"T. Essel" (hiding somewhere in cyberspace)\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>I complained:
> > [...] as far as I know, there is currently -no- textbook on "How to
> > Theorize from a Lagrangian"
On Fri, 2 Apr 2004, MM replied:
> I would have said that Peskin & Schroeder's "Introduction to QFT", and
> Weinberg's multi-volume "The Quantum Theory of Fields" together do a
> good job.
OK, I've made a mental note. But AFAIK neither of these two study the
questions I asked regarding -classical- field theories. Does anyone else
agree that those questions are by no means trivial? I am beginning to
wonder...
I trust people who have the books I mentioned, namely
1. Landau/Lifschitz, Mechanics,
2. Landau/Lifschitz, Classical Theory of Fields),
3. Ohanian/Ruffini, Gravitation and Spacetime,
4. Weinberg, Gravitation and Cosmology
will agree that the questions I raised are largely unaddressed in these
books, although from playing around I do understand why LL and OR make
certain comments which only apply in examples beyond the two examples
these books discuss.
I expect (perhaps) naively that my questions should also be relevant in
QFT, e.g. because I am vaguely aware of mathematical analogies between the
usual suspects (PDEs arising in classical field theory and in quantum
theory). Is this expectation also wrong?
"T. Essel" (hiding somewhere in cyberspace)
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>Perhaps the thing to do is to just write down the most general\nlagrangian you can think off that describes the standard model and\ngeneral relativity. Eg the thing that is experimentally the accepted\nnorm.\n\nPalatini and others go into great length to do just that, using the\ntechnology of fiber bundles and differential forms.\n\nIts messy, and not the best way to learn, but there you have it... THE\nLagrangian of physics so to speak!\n\nA few general concepts are of course applicable. The first is that,\nby and large the lagrangian are constructed using symmetry principles.\n\nThe second is that gravity does indeed behave a little differently, as\nit involves metric forms as the fundamental thing, as opposed to guage\nconnections.\n\nThe third, is how to mix quantum mechanics into the picture. In\nessense, its how you interpret the lagrangian that changes from the\nmove to classical systems to quantum mechanics. Of course,\nrenormalizability then cuts down on the total amount of terms, and\njustifies why certain things were left out of the lagrangian that a\npriori could have been there in the first place.\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>Perhaps the thing to do is to just write down the most general
lagrangian you can think off that describes the standard model and
general relativity. Eg the thing that is experimentally the accepted
norm.
Palatini and others go into great length to do just that, using the
technology of fiber bundles and differential forms.
Its messy, and not the best way to learn, but there you have it... THE
Lagrangian of physics so to speak!
A few general concepts are of course applicable. The first is that,
by and large the lagrangian are constructed using symmetry principles.
The second is that gravity does indeed behave a little differently, as
it involves metric forms as the fundamental thing, as opposed to guage
connections.
The third, is how to mix quantum mechanics into the picture. In
essense, its how you interpret the lagrangian that changes from the
move to classical systems to quantum mechanics. Of course,
renormalizability then cuts down on the total amount of terms, and
justifies why certain things were left out of the lagrangian that a
priori could have been there in the first place.
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>on 05/04/2004 9:54 am, tessel@um.bot at tessel@um.bot wrote:\n\n> On Sun, 4 Apr 2004, Aaron Bergman wrote:\n>\n>> All the issues you raise are pretty scattered.\n>\n> Well, do you at least agree there\'s some point to gathering the relevant\n> ideas together in one place?\n>\n>> Goldstein (even though I didn\'t use it) almost assuredly covers\n>> Lagrangians quite extensively.\n>\n> Fine, but how many of the specific "issues for discussion" which I\n> mentioned does this book cover? Also, I think I probably know which book\n> you have in mind, but can you please give a citation?\n>\n> (Sorry, I still haven\'t gotten around to reskimming Peskin and Shroeder,\n> but I promise to do this.)\n>\n>> It\'s not all done in one place in the curriculum always, but I haven\'t\n>> really noticed a gap.\n>\n> Many thanks to Frank Hellman, who commented that he agrees that there -is-\n> something of a gap, and gave another example of the kind of [serious]\n> "Lagranian play" I had in mind. I also eagerly await any comments by\n> Steve Carlip, e.g. regarding searching for an EIH analog.\n\nI agree too, I\'ve never seen anything remotely like what you describe as\nyour ideal. All the introductions to Lagrangians in mechanics textbooks that\nI have seen, are very skimpy in comparison.\n\nTim\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>on 05/04/2004 9:54 am, tessel@um.bot at tessel@um.bot wrote:
> On Sun, 4 Apr 2004, Aaron Bergman wrote:
>
>> All the issues you raise are pretty scattered.
>
> Well, do you at least agree there's some point to gathering the relevant
> ideas together in one place?
>
>> Goldstein (even though I didn't use it) almost assuredly covers
>> Lagrangians quite extensively.
>
> Fine, but how many of the specific "issues for discussion" which I
> mentioned does this book cover? Also, I think I probably know which book
> you have in mind, but can you please give a citation?
>
> (Sorry, I still haven't gotten around to reskimming Peskin and Shroeder,
> but I promise to do this.)
>
>> It's not all done in one place in the curriculum always, but I haven't
>> really noticed a gap.
>
> Many thanks to Frank Hellman, who commented that he agrees that there -is-
> something of a gap, and gave another example of the kind of [serious]
> "Lagranian play" I had in mind. I also eagerly await any comments by
> Steve Carlip, e.g. regarding searching for an EIH analog.
I agree too, I've never seen anything remotely like what you describe as
your ideal. All the introductions to Lagrangians in mechanics textbooks that
I have seen, are very skimpy in comparison.
Tim
Aaron Bergman
Apr7-04, 08:59 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>In article <40711eba$1@news.sentex.net>, tessel@um.bot wrote:\n\n> On Sun, 4 Apr 2004, Aaron Bergman wrote:\n>\n> > All the issues you raise are pretty scattered.\n>\n> Well, do you at least agree there\'s some point to gathering the relevant\n> ideas together in one place?\n\nThis is generally done in math textbooks, although I wouldn\'t be\nsurprised if there were physics textbooks out there that did the same.\n>\n> > Goldstein (even though I didn\'t use it) almost assuredly covers\n> > Lagrangians quite extensively.\n>\n> Fine, but how many of the specific "issues for discussion" which I\n> mentioned does this book cover? Also, I think I probably know which book\n> you have in mind, but can you please give a citation?\n\nH. Goldstein _Classical Mechanics_. I don\'t own a copy, so I can\'t tell\nyou precisely what\'s in it, but it\'s one of the standard junior level\nclassical mechanics texts.\n\nAaron\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>In article <40711eba$1@news.sentex.net>, tessel@um.bot wrote:
> On Sun, 4 Apr 2004, Aaron Bergman wrote:
>
> > All the issues you raise are pretty scattered.
>
> Well, do you at least agree there's some point to gathering the relevant
> ideas together in one place?
This is generally done in math textbooks, although I wouldn't be
surprised if there were physics textbooks out there that did the same.
>
> > Goldstein (even though I didn't use it) almost assuredly covers
> > Lagrangians quite extensively.
>
> Fine, but how many of the specific "issues for discussion" which I
> mentioned does this book cover? Also, I think I probably know which book
> you have in mind, but can you please give a citation?
H. Goldstein _Classical Mechanics_. I don't own a copy, so I can't tell
you precisely what's in it, but it's one of the standard junior level
classical mechanics texts.
Aaron
tessel@um.bot
Apr7-04, 09:10 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>On Fri, 2 Apr 2004, Danny Ross Lunsford wrote:\n\n> tessel@um.bot wrote:\n>\n> > Nonetheless, as far as I know, there is currently -no- textbook on\n> > "How to Theorize from a Lagrangian"! (But please correct me, with\n> > citations, if I am wrong about this.)\n>\n> As far as I know this matter is well-covered if scattered in the\n> literature. The key paper for the physicist is by Belinfante and\n> concerns the issue of symmetrizing the canonical energy tensor.\n\nOh good, Ohanian/Ruffini make the requisite remark but fail to give any\ncitation. Can you or anyone else give me a bit more than just an author\nname?\n\n> To this one can add the work of Palatini where he shows how to get GR by\n> varying the connection.\n>\n> For textbooks, one can mention Landau-Lifschitz and Barut.\n^^^^^^^^^^^^^^^^\n\nSigh... by now it seems clear that my original post must have been very\nbadly worded, and I apologize!\n\nI did specifically mention (but I should have given a list)\n\nLL, Mechanics,\n\nLL, Classical Theory of Fields,\n\nOhanian/Ruffini, Gravitation and Spacetime,\n\nWeinberg, Gravitation and Cosmology\n\nas textbooks I myself have studied. I regard LL as superb on the topics\ncovered; my claim was that this coverage is -inadequate- the purpose I\nhave in mind. (I think I understand LL, but another subsidiary point here\nis that in my experience, the average student will not. As evidence that\nI may not be the only one who thinks LL/Weinberg/MTW aim too high for many\nstudents here, I note the steady decrease in the mathematical\nsophistication assumed by authors from the "first generation" classics,\nLL/Weinberg/MTW, through the excellent "second generation" textbooks by\nSchutz/D\'Inverno, to the recent "third generation" textbooks by Carroll\nand Hartle.\n\n(*Heh---since Carroll apparently sometimes reads this group, maybe he will\ncomment on this alleged downdumbing? BTW, despite this comment, I think\nthis is an excellent textbook; see RWWW! Downdumbing isn\'t neccessarily a\nbad thing--- in a way, what I am doing in this thread is proposing to\nexpand some side remarks by LL/OR into a sketch of the textbook length\ndiscussion I feel these issues deserve, precisely because, while subtle,\nthey seem to frequently arise in the -practice- of field theorizing.)\n\nWhat purpose do I mean? Well, to repeat my list of "topics for\ndiscussion" and my example of one possible "stated goal" for a field\ntheory defined in terms of a Lagrangian (and possibly additional\nassumptions, e.g. if needed, a force law):\n\n======= BEGIN SELF QUOTATION ============\n\nI propose to start as simply as possible and slowly increase the level of\nsophistication, e.g. (not neccessarily in this order)\n\n1. first linear, then nonlinear Lagrangians,\n\n2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more\nvariables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),\n\n3. first scalar theories, then vector theories, then tensor theories,\n\n4. first classical field theories, then quantum field theories.\n\n[5. first theories admitting U(1) gauge symmetries, then nonabelian gauge\nsymmetries.]\n\nIssues we might systematically explore in this way could include:\n\n1. how to sensibly select the terms of our Lagrangian, as written in the\nuseful form\n\nL = L_matt + L_interaction + L_field\n\n2. how to guess key features of the "free-field" case of the field\nequation of our theory from looking over L_field,\n\n3. how to write down the Euler-Lagrange equations (allowing for the\npossibility of second and higher order derivatives),\n\n4. how to verify key features of the (free-field) field equation, e.g.\n\n(a) how to determine the group of "point symmetries" and the subgroup of\n"variational symmetries",\n\n(b) how to confirm the existence of wavelike solutions to the (free-field)\nfield equation,\n\n(c) how to obtain and study particular solutions (even for nonlinear PDEs)\ne.g. using Lie\'s methods, Baecklund symmetries, scattering transform,\n\n5. how to obtain a suitable energy-momentum tensor (and what to do if the\n"canonical energy-momentum tensor" turns out to be -nonsymmetric-),\n\n6. more generally how to find "conserved currents" (e.g. from "divergence\nsymmetries"), at least in the case where L only contains first order\nderivatives,\n\n7. how to find suitable "conserved currents" if L contains second order\nderivatives,\n\n8. how to determine L_interaction from energy-momentum exchange between\nthe field and "particles", if this is possible, or if not, how otherwise\nto obtain a suitable "equation of motion" for "charged" particles,\n\n9. how to check for at least the most commonly encountered\nself-inconsistencies which can arise in a field theory,\n\nHaving explored these issues for a generous range of simple toy theories,\nwe can try to address the question which presumably we would all agree is\nthe question of ultimate interest: how can we write down a Lagrangian\nwhich has a reasonable chance of producing a theory which will accomplish\nstated goals?\n\nLet me mention a specific example of a "stated goal". Recall that\nMaxwell\'s theory of EM is beset with singularities in the field. Recall\nalso that a few years ago we had some people here who wanted to replace\ngtr with a theory which would "ameliorate" singularities while preserving\nthe remarkable experimental/observational success of that theory. In\nprinciple, as I said at the time, that\'s not a bad idea, but I claimed\nthen (and repeat now) that this is a -much- more challenging program than\n(I felt) those people recognized. At that time I proposed to step back and\nconsider first the analogous but easier question for EM, but unfortunately\n(as I recall the affair, at least) my correspondents were insufficiently\nwilling to do this. I hope and believe that the current student\ngeneration is more amenable to such good advice!\n\nActually, I propose to start with scalar theories, e.g.\nthree-dimensional analogues of\n\nL = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ]\n\n(I might have munged a sign or two, but this is supposed to be the\nLagrangian which leads to the sine-Gordon equation, a nonlinear wave\nequation beloved of PDE people), or more generally\n\nL = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term"\nwhich is small for small u\n\n(since I doubt the sine-Gordon equation will fully achieve the goal of\n"ameliorizating" singularities in our scalar field).\n\n======= END SELF QUOTATION =============\n\nI doubt that anyone familiar with the books I listed just above (at least\nnot anyone who has actually tried to play with Lagrangians) would say that\nthey are adequate for working toward the "stated goal" I mentioned as one\npossible example! Of course, an excellent student should in principle be\nable to figure out what is needed on his/her own. But most students\n(obviously!) are average, not excellent. I was in effect suggesting that\nthe better students--- I immodestly implicitly counted myself in that\ngroup!--- help out the less able by illustrating "intelligent field\ntheorizing", by working through the detailed analysis of some interesting\ntoy theories.\n\nAgain, I was not saying that existing textbooks do not cover Lagrangians,\nbut rather than for intelligently positing/analyzing even toy theories,\nconsiderably more background material/insight is needed. Apparently this\nbackground material has not yet been presented in a single textbook,\nwhich, I said, I find very odd, even though I mentioned a plausible reason\nwhy this might be the case.\n\nPlease note that I was also saying that part of the point here would be to\nguide average students in the very first step, the -intelligent- choice of\na Lagrangian to analyze. And the last step, how to modify that initial\nguess with the benefit of the experience gained by analyzing the alleged\nfield theory it "defines" (perhaps with additional stuff about "force\nlaw"--- see my comment about the nontriviality of EIH).\n\nA major point of my proposal to fill an alleged gap was this: only by\nplaying with a goodly number of toy theories can one begin to appreciate\nwhat is likely to be a "bad" choice of Lagrangian. C.f. Doug\'s proposal\nin the "Weinberg on GR" thread, which I regard as "obviously bad".\n(Doug---sorry if I seem to picking on you; I\'m not; I already noted that\nwhile I think you munged it in your choice of Lagrangian and what you did\nthereafer, but I said that I don\'t think this is your fault, since\nexisting textbooks give so little guidance for the -practice- of field\ntheorizing) Recall also that as far as I can tell, Doug has studied\npretty much the same books that I have.\n\n> Of course one should have a good understanding of variational calculus\n> going in. For the physicist, one can mention Caratheordory and\n> Gelfand-Fomin. I remember the book by Weinstock as being loaded with\n> excellent problem sets.\n\nIIRC, Weinstock does mention the more general form of the Euler-Lagrange\nequation. Unfortunately, I don\'t have Gelfand-Fomin, but Caratheodory is\na classic!\n\n> If you are interested in seeing the process at work in rather new and\n> non-trivial circumstances, drop me an email.\n\nGo for it! (I.e., an expository post to this n.g.--- but recall my own\nregretable but undeniable lack of background in quantum physics.)\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Fri, 2 Apr 2004, Danny Ross Lunsford wrote:
> tessel@um.bot wrote:
>
> > Nonetheless, as far as I know, there is currently -no- textbook on
> > "How to Theorize from a Lagrangian"! (But please correct me, with
> > citations, if I am wrong about this.)
>
> As far as I know this matter is well-covered if scattered in the
> literature. The key paper for the physicist is by Belinfante and
> concerns the issue of symmetrizing the canonical energy tensor.
Oh good, Ohanian/Ruffini make the requisite remark but fail to give any
citation. Can you or anyone else give me a bit more than just an author
name?
> To this one can add the work of Palatini where he shows how to get GR by
> varying the connection.
>
> For textbooks, one can mention Landau-Lifschitz and Barut.
^^^^^^^^^^^^^^^^
Sigh... by now it seems clear that my original post must have been very
badly worded, and I apologize!
I did specifically mention (but I should have given a list)
LL, Mechanics,
LL, Classical Theory of Fields,
Ohanian/Ruffini, Gravitation and Spacetime,
Weinberg, Gravitation and Cosmology
as textbooks I myself have studied. I regard LL as superb on the topics
covered; my claim was that this coverage is -inadequate- the purpose I
have in mind. (I think I understand LL, but another subsidiary point here
is that in my experience, the average student will not. As evidence that
I may not be the only one who thinks LL/Weinberg/MTW aim too high for many
students here, I note the steady decrease in the mathematical
sophistication assumed by authors from the "first generation" classics,
LL/Weinberg/MTW, through the excellent "second generation" textbooks by
Schutz/D'Inverno, to the recent "third generation" textbooks by Carroll
and Hartle.
(*Heh---since Carroll apparently sometimes reads this group, maybe he will
comment on this alleged downdumbing? BTW, despite this comment, I think
this is an excellent textbook; see RWWW! Downdumbing isn't neccessarily a
bad thing--- in a way, what I am doing in this thread is proposing to
expand some side remarks by LL/OR into a sketch of the textbook length
discussion I feel these issues deserve, precisely because, while subtle,
they seem to frequently arise in the -practice- of field theorizing.)
What purpose do I mean? Well, to repeat my list of "topics for
discussion" and my example of one possible "stated goal" for a field
theory defined in terms of a Lagrangian (and possibly additional
assumptions, e.g. if needed, a force law):
======= BEGIN SELF QUOTATION ============
I propose to start as simply as possible and slowly increase the level of
sophistication, e.g. (not neccessarily in this order)
1. first linear, then nonlinear Lagrangians,
2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more
variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),
3. first scalar theories, then vector theories, then tensor theories,
4. first classical field theories, then quantum field theories.
[5. first theories admitting U(1) gauge symmetries, then nonabelian gauge
symmetries.]
Issues we might systematically explore in this way could include:
1. how to sensibly select the terms of our Lagrangian, as written in the
useful form
L = L_{matt} + L_{interaction} + L_{field}
2. how to guess key features of the "free-field" case of the field
equation of our theory from looking over L_{field},
3. how to write down the Euler-Lagrange equations (allowing for the
possibility of second and higher order derivatives),
4. how to verify key features of the (free-field) field equation, e.g.
(a) how to determine the group of "point symmetries" and the subgroup of
"variational symmetries",
(b) how to confirm the existence of wavelike solutions to the (free-field)
field equation,
(c) how to obtain and study particular solutions (even for nonlinear PDEs)
e.g. using Lie's methods, Baecklund symmetries, scattering transform,
5. how to obtain a suitable energy-momentum tensor (and what to do if the
"canonical energy-momentum tensor" turns out to be -nonsymmetric-),
6. more generally how to find "conserved currents" (e.g. from "divergence
symmetries"), at least in the case where L only contains first order
derivatives,
7. how to find suitable "conserved currents" if L contains second order
derivatives,
8. how to determine L_{interaction} from energy-momentum exchange between
the field and "particles", if this is possible, or if not, how otherwise
to obtain a suitable "equation of motion" for "charged" particles,
9. how to check for at least the most commonly encountered
self-inconsistencies which can arise in a field theory,
Having explored these issues for a generous range of simple toy theories,
we can try to address the question which presumably we would all agree is
the question of ultimate interest: how can we write down a Lagrangian
which has a reasonable chance of producing a theory which will accomplish
stated goals?
Let me mention a specific example of a "stated goal". Recall that
Maxwell's theory of EM is beset with singularities in the field. Recall
also that a few years ago we had some people here who wanted to replace
gtr with a theory which would "ameliorate" singularities while preserving
the remarkable experimental/observational success of that theory. In
principle, as I said at the time, that's not a bad idea, but I claimed
then (and repeat now) that this is a -much- more challenging program than
(I felt) those people recognized. At that time I proposed to step back and
consider first the analogous but easier question for EM, but unfortunately
(as I recall the affair, at least) my correspondents were insufficiently
willing to do this. I hope and believe that the current student
generation is more amenable to such good advice!
Actually, I propose to start with scalar theories, e.g.
three-dimensional analogues of
L = -1/2 [ u_t^2 - u_x^2 - m^2 cos(u)^2 ]
(I might have munged a sign or two, but this is supposed to be the
Lagrangian which leads to the sine-Gordon equation, a nonlinear wave
equation beloved of PDE people), or more generally
L = -1/2 [ u_t^2 - u_x^2 ] + some other proposed "perturbing term"
which is small for small u
(since I doubt the sine-Gordon equation will fully achieve the goal of
"ameliorizating" singularities in our scalar field).
======= END SELF QUOTATION =============
I doubt that anyone familiar with the books I listed just above (at least
not anyone who has actually tried to play with Lagrangians) would say that
they are adequate for working toward the "stated goal" I mentioned as one
possible example! Of course, an excellent student should in principle be
able to figure out what is needed on his/her own. But most students
(obviously!) are average, not excellent. I was in effect suggesting that
the better students--- I immodestly implicitly counted myself in that
group!--- help out the less able by illustrating "intelligent field
theorizing", by working through the detailed analysis of some interesting
toy theories.
Again, I was not saying that existing textbooks do not cover Lagrangians,
but rather than for intelligently positing/analyzing even toy theories,
considerably more background material/insight is needed. Apparently this
background material has not yet been presented in a single textbook,
which, I said, I find very odd, even though I mentioned a plausible reason
why this might be the case.
Please note that I was also saying that part of the point here would be to
guide average students in the very first step, the -intelligent- choice of
a Lagrangian to analyze. And the last step, how to modify that initial
guess with the benefit of the experience gained by analyzing the alleged
field theory it "defines" (perhaps with additional stuff about "force
law"--- see my comment about the nontriviality of EIH).
A major point of my proposal to fill an alleged gap was this: only by
playing with a goodly number of toy theories can one begin to appreciate
what is likely to be a "bad" choice of Lagrangian. C.f. Doug's proposal
in the "Weinberg on GR" thread, which I regard as "obviously bad".
(Doug---sorry if I seem to picking on you; I'm not; I already noted that
while I think you munged it in your choice of Lagrangian and what you did
thereafer, but I said that I don't think this is your fault, since
existing textbooks give so little guidance for the -practice- of field
theorizing) Recall also that as far as I can tell, Doug has studied
pretty much the same books that I have.
> Of course one should have a good understanding of variational calculus
> going in. For the physicist, one can mention Caratheordory and
> Gelfand-Fomin. I remember the book by Weinstock as being loaded with
> excellent problem sets.
IIRC, Weinstock does mention the more general form of the Euler-Lagrange
equation. Unfortunately, I don't have Gelfand-Fomin, but Caratheodory is
a classic!
> If you are interested in seeing the process at work in rather new and
> non-trivial circumstances, drop me an email.
Go for it! (I.e., an expository post to this n.g.--- but recall my own
regretable but undeniable lack of background in quantum physics.)
"T. Essel" (hiding somewhere in cyberspace)
Danny Ross Lunsford
Apr7-04, 09:13 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>tessel@um.bot wrote:\n\n[..about teaching and learning..]\n\nLike many here I learned a lot of things on my own when the time came.\nBecause I was happy reading older books (esp. Klein) I was used to doing\nthings in terms of parameters, including differentiating with respect to\nthem (for determining the envelope). So "differentiating this thing with\nrespect to the amplitude" or the like seemed very natural, and I didn\'t\nget hung up on taking derivatives with respect to a velocity. (I think\nit helps to go directly to the Hamiltonian formulation. Many people get\nall balled up trying to imagine d/dxdot.) Also I was very comfortable\nwith "first-order thinking" - calculating with differentials, because of\nthe way I learned calculus and DEs. I was accidentally well-prepared.\n\nI think I really learned what was going on from Sommerfeld. His little\nmechanics book is outstanding. See the sections on "Principle of Virtual\nWork".\n\n-drl\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>tessel@um.bot wrote:
[..about teaching and learning..]
Like many here I learned a lot of things on my own when the time came.
Because I was happy reading older books (esp. Klein) I was used to doing
things in terms of parameters, including differentiating with respect to
them (for determining the envelope). So "differentiating this thing with
respect to the amplitude" or the like seemed very natural, and I didn't
get hung up on taking derivatives with respect to a velocity. (I think
it helps to go directly to the Hamiltonian formulation. Many people get
all balled up trying to imagine d/dxdot.) Also I was very comfortable
with "first-order thinking" - calculating with differentials, because of
the way I learned calculus and DEs. I was accidentally well-prepared.
I think I really learned what was going on from Sommerfeld. His little
mechanics book is outstanding. See the sections on "Principle of Virtual
Work".
-drl
Aaron Bergman
Apr7-04, 09:13 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resize=yes,status=no,wi dth=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>In article <40711eca\\$1@news.sentex.net>, tessel@um.bot wrote:\n\n> Or again: when I proposed to try to derive the equation of motion of\n> "charged" test particles in a given field, did you and Aaron fail to\n> notice that I was talking about the analogue of the EIH procedure in the\n> case of gtr? (For other readers: Einstein, Infeld, and Hoffmann proposed\n> to remove the assumption that the world lines nonspinning test particles\n> in a vacuum region are timelike geodesics by -deriving- this from the\n> Einstein field equation.) I assume that you and Aaron are both aware that\n> this is -highly nontrivial-; if not, see\n\nUnless I misunderstand you, this is a standard exercise in many GR\nbooks. It\'s done in chapter 9 of Stephani, for example. It\'s not\nparticularly difficult either, so maybe you can elaborate further about\nwhat you are referring to?\n\n[...]\n\n> I wonder if either Shagird or Aaron really think that average students\n> will be able to attack successfully the example of "stated goal" which I\n> mentioned without help well beyond that offered by Landau/Lifshitz or\n> Ohanian/Rufinni? Or the question of searching for an analogue to the\n> proposal of EIH? Or even searching for "wave solutions" to a nonlinear\n> free-field equation like the sine-Gordon equation?\n\nDepends on the student. It sounds like what you really want is an\nadvanced course in PDEs, in particular, those stemming from lagrangians.\nI wouldn\'t call this a gap, nonetheless, because I don\'t see too many\nproblems stemming from the lack of what you\'re looking for. I\'m not\nsurprised you can point to bad papers on the ArXiv. There are papers\nwith misunderstandings of more things than just PDEs there. There\'s a\nlot more things one could add to the curriculum, but aren\'t there. A\ngood student should be able to pick up a textbook and learn additional\nstuff as needed.\n\nAaron\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>In article <40711eca$1@news.sentex.net>, tessel@um.bot wrote:
> Or again: when I proposed to try to derive the equation of motion of
> "charged" test particles in a given field, did you and Aaron fail to
> notice that I was talking about the analogue of the EIH procedure in the
> case of gtr? (For other readers: Einstein, Infeld, and Hoffmann proposed
> to remove the assumption that the world lines nonspinning test particles
> in a vacuum region are timelike geodesics by -deriving- this from the
> Einstein field equation.) I assume that you and Aaron are both aware that
> this is -highly nontrivial-; if not, see
Unless I misunderstand you, this is a standard exercise in many GR
books. It's done in chapter 9 of Stephani, for example. It's not
particularly difficult either, so maybe you can elaborate further about
what you are referring to?
[...]
> I wonder if either Shagird or Aaron really think that average students
> will be able to attack successfully the example of "stated goal" which I
> mentioned without help well beyond that offered by Landau/Lifshitz or
> Ohanian/Rufinni? Or the question of searching for an analogue to the
> proposal of EIH? Or even searching for "wave solutions" to a nonlinear
> free-field equation like the sine-Gordon equation?
Depends on the student. It sounds like what you really want is an
advanced course in PDEs, in particular, those stemming from lagrangians.
I wouldn't call this a gap, nonetheless, because I don't see too many
problems stemming from the lack of what you're looking for. I'm not
surprised you can point to bad papers on the ArXiv. There are papers
with misunderstandings of more things than just PDEs there. There's a
lot more things one could add to the curriculum, but aren't there. A
good student should be able to pick up a textbook and learn additional
stuff as needed.
Aaron
Buzurg Shagird
Apr7-04, 09:25 AM
<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>\n\n\ntessel@um.bot wrote in message news:<40711eca\\$1@news.sentex.net>...\n\n> On Sun, 4 Apr 2004, Buzurg Shagird wrote:\n>\n> > I am not sure where you see that gap ... surely Landau and Lifshitz do a\n> > fairly good job of discussing the action principle in their Classical\n> > Mechanics and Classical Field Theory books?\n>\n> Since I mentioned those books as inadequate for students like Doug\n> Sweetser who are trying to formulate and play around with a "Lagrangian\n> field theory" (even a "toy theory"), you may assume that I feel they (and\n> the other books I mentioned) do -not- explain -the "issues for discussion"\n> which I listed- (see below for a repeat of this list).\n\nWell, first of all, when I said that I wasn\'t sure where you see that\ngap, I did NOT mean ``I know everything there is to know about field\ntheory", NOR ``all possible issues of classical field theory has been\ndiscussed in those books." If you got that impression from my post, I\nam really sorry, that was not my intent at all.\n\nI am sure you will agree that no textbook can discuss every possible\nissue in a given subject, then it will not be a textbook! A textbook\nis meant for creating a foundation in a given subject, usually by\nmeans of a `linearly progressive\' discussion from some starting point\nto some end point, plus occasionally a brief discussion in some\ntangential topics which the author finds interesting. A good textbook\nis read by a lot of students, many of whom will not carry on to do\nresearch in the subject, but will find an overview quite useful.\n\nYour list of ``issues for discussion" includes some things which are\ncovered in the standard textbooks, some things which are covered in\nthe non-standard textbooks, i.e. books which compile existing\nknowledge, but with a narrow focus or from an uncommon viewpoint, and\na few things for which you have look in monographs or research papers.\nBut this kind of situation will exist whenever you want to make a plan\nfor a thorough investigation of some topic, as you have done.\n\nI am unable to see the `gap\' that you mention because I feel that\nthose books prepare you well enough to do your own investigations into\nthe questions that you have asked -- and as far as I could see from\nyour post, you are indeed well prepared to do that investigation. That\nis why I was surprised that you see a gap, or that you expect the\nanswers to all your questions in some textbook.\n\n> My claim is quite different: for purposes of playing around intelligently\n> with Lagrangians, -much more is needed-.\n\nOnly by those who want to `play around\' with Lagrangians, in the way\nthat you and a few others on this group may want to do. :-) But that\nis very much a `special interests\' topic. Why would a textbook include\nall possible ways of playing around with Lagrangians?\n\nInstead, what good textbooks do is to point out the _principles_\nbehind the most common Lagrangians, so that you can make your own\nphysical or mathematical principle, your own action, and your own\nperturbative quantum field theory (and if you are really lucky,\nyour own non-perturbative qft). :-)\n\n> > You could complain that Dirac\'s method of analysis of constraints is not\n> > given in more books,\n>\n> I didn\'t mention that, probably because I haven\'t heard of it, at least\n> not under that name.\n\nYou may find it useful to learn about it -- I was being somewhat\nfacetious when I said what\'s below. You can learn a lot about a field\ntheory by studying the structure of its constraints -- in particular\nyou can identify the physical and unphysical degrees of freedom, and\nhave a fairly good idea of how symmetries are implemented on the\nquantum states ...\n\n>\n> > but you can\'t apply it directly to non-Abelian gauge theories anyway ---\n> > you need to go to superspace or such to introduce ghosts, and ghosts are\n> > better done in the path integral formalism anyway -- so you shouldn\'t\n> > really complain. :-)\n\n> I wonder if either Shagird or Aaron really think that average students\n> will be able to attack successfully the example of "stated goal" which I\n> mentioned without help well beyond that offered by Landau/Lifshitz or\n\nI can\'t speak for Aaron, but I certainly don\'t expect a student,\naverage or not, to attack any major research problem immediately after\nreading a standard textbook. I expect them to skim through many books,\nmongraphs and papers looking for hints and discussions, and above all,\nto talk to people who are interested in the same sort of problems.\n\n-S.\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>tessel@um.bot wrote in message news:<40711eca$1@news.sentex.net>...
> On Sun, 4 Apr 2004, Buzurg Shagird wrote:
>
> > I am not sure where you see that gap ... surely Landau and Lifshitz do a
> > fairly good job of discussing the action principle in their Classical
> > Mechanics and Classical Field Theory books?
>
> Since I mentioned those books as inadequate for students like Doug
> Sweetser who are trying to formulate and play around with a "Lagrangian
> field theory" (even a "toy theory"), you may assume that I feel they (and
> the other books I mentioned) do -not- explain -the "issues for discussion"
> which I listed- (see below for a repeat of this list).
Well, first of all, when I said that I wasn't sure where you see that
gap, I did NOT mean ``I know everything there is to know about field
theory", NOR ``all possible issues of classical field theory has been
discussed in those books." If you got that impression from my post, I
am really sorry, that was not my intent at all.
I am sure you will agree that no textbook can discuss every possible
issue in a given subject, then it will not be a textbook! A textbook
is meant for creating a foundation in a given subject, usually by
means of a `linearly progressive' discussion from some starting point
to some end point, plus occasionally a brief discussion in some
tangential topics which the author finds interesting. A good textbook
is read by a lot of students, many of whom will not carry on to do
research in the subject, but will find an overview quite useful.
Your list of ``issues for discussion" includes some things which are
covered in the standard textbooks, some things which are covered in
the non-standard textbooks, i.e. books which compile existing
knowledge, but with a narrow focus or from an uncommon viewpoint, and
a few things for which you have look in monographs or research papers.
But this kind of situation will exist whenever you want to make a plan
for a thorough investigation of some topic, as you have done.
I am unable to see the `gap' that you mention because I feel that
those books prepare you well enough to do your own investigations into
the questions that you have asked -- and as far as I could see from
your post, you are indeed well prepared to do that investigation. That
is why I was surprised that you see a gap, or that you expect the
answers to all your questions in some textbook.
> My claim is quite different: for purposes of playing around intelligently
> with Lagrangians, -much more is needed-.
Only by those who want to `play around' with Lagrangians, in the way
that you and a few others on this group may want to do. :-) But that
is very much a `special interests' topic. Why would a textbook include
all possible ways of playing around with Lagrangians?
Instead, what good textbooks do is to point out the _principles_
behind the most common Lagrangians, so that you can make your own
physical or mathematical principle, your own action, and your own
perturbative quantum field theory (and if you are really lucky,
your own non-perturbative qft). :-)
> > You could complain that Dirac's method of analysis of constraints is not
> > given in more books,
>
> I didn't mention that, probably because I haven't heard of it, at least
> not under that name.
You may find it useful to learn about it -- I was being somewhat
facetious when I said what's below. You can learn a lot about a field
theory by studying the structure of its constraints -- in particular
you can identify the physical and unphysical degrees of freedom, and
have a fairly good idea of how symmetries are implemented on the
quantum states ...
>
> > but you can't apply it directly to non-Abelian gauge theories anyway ---
> > you need to go to superspace or such to introduce ghosts, and ghosts are
> > better done in the path integral formalism anyway -- so you shouldn't
> > really complain. :-)
> I wonder if either Shagird or Aaron really think that average students
> will be able to attack successfully the example of "stated goal" which I
> mentioned without help well beyond that offered by Landau/Lifshitz or
I can't speak for Aaron, but I certainly don't expect a student,
average or not, to attack any major research problem immediately after
reading a standard textbook. I expect them to skim through many books,
mongraphs and papers looking for hints and discussions, and above all,
to talk to people who are interested in the same sort of problems.
-S.
Doug Sweetser
Apr9-04, 04:17 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>Hello T. Essel:\n\nI was in the position four years ago of having field equations, but no\nLagrangian. What I did was practice with EM, doing the derivation from\nthe classical EM Lagrange density out to the field equations about a\ndozen times. In many ways, it is like practicing scales or\nmultiplication tables: nothing can substitute for practice. After\nthat, it was possible for me to play with variations.\n\nWorking out the units gave me a great insight. A Lagrange density is\nall the mass (aka energy) in a volume. This is a scalar that has\neverything that can happen in a box. The action integrates over the\nvolume and time. If that integral does not change as something is\nvaried over an arbitrary amount of time, then a symmetry is found. [In\na personal language observation, it is why I prefer the phrase\n"Lagrange density" to "Lagrangian" because it reminds me of all the\nenergy in a volume.]\n\nThe field I was trying to find a Lagrange density for was this:\n\nJq^u - Jm^u = A^u;v_;v\n\nwhich looks quite similar to EM:\n\nJq^u = A^u,v_,v\n\nTherefore it was appropriate to try variations, once simple repetition\nlead to some practical, not linguistic, mastery. The Lagrange density\nI study is this one:\n\nL = -(Jq^u - Jm^u) A_u - 1/2 A^u;v A_u;v\n\nThe first time I wrote this down, I had Jm^u with the same sign as\nJq^u. That leads to fields were all like charges repel. The signs\nbetween the coupling term (J^u A_u) and the field strength term (A^u;v\nA_u;v) determine if like charges attract or repel. It would have\nprevented some embarrassment for me if I had read that first :-)\n\nI also had quite a bit of fear about calculating the stress energy\ntensor. Again, practice with EM removed that inhibition. I was able\nto do that for this Lagrange density.\n\nBefore I ramble about anything else, I request you take the above\nLagrange density and generate those field equations. Practice is good!\nIf one is skilled with an action, then it turns out to be a one liner.\nThe way I did it, I wrote out all the terms without indices, then\nstarted to take derivatives. That process takes a few pages.\n\n\ndoug\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>Hello T. Essel:
I was in the position four years ago of having field equations, but no
Lagrangian. What I did was practice with EM, doing the derivation from
the classical EM Lagrange density out to the field equations about a
dozen times. In many ways, it is like practicing scales or
multiplication tables: nothing can substitute for practice. After
that, it was possible for me to play with variations.
Working out the units gave me a great insight. A Lagrange density is
all the mass (aka energy) in a volume. This is a scalar that has
everything that can happen in a box. The action integrates over the
volume and time. If that integral does not change as something is
varied over an arbitrary amount of time, then a symmetry is found. [In
a personal language observation, it is why I prefer the phrase
"Lagrange density" to "Lagrangian" because it reminds me of all the
energy in a volume.]
The field I was trying to find a Lagrange density for was this:
Jq^u - Jm^u = A^u;v_;v
which looks quite similar to EM:
Jq^u = A^u,v_,v
Therefore it was appropriate to try variations, once simple repetition
lead to some practical, not linguistic, mastery. The Lagrange density
I study is this one:
L = -(Jq^u - Jm^u) A_u - 1/2 A^u;v A_u;v
The first time I wrote this down, I had Jm^u with the same sign as
Jq^u. That leads to fields were all like charges repel. The signs
between the coupling term (J^u A_u) and the field strength term (A^u;v
A_u;v) determine if like charges attract or repel. It would have
prevented some embarrassment for me if I had read that first :-)
I also had quite a bit of fear about calculating the stress energy
tensor. Again, practice with EM removed that inhibition. I was able
to do that for this Lagrange density.
Before I ramble about anything else, I request you take the above
Lagrange density and generate those field equations. Practice is good!
If one is skilled with an action, then it turns out to be a one liner.
The way I did it, I wrote out all the terms without indices, then
started to take derivatives. That process takes a few pages.
doug
Igor Khavkine
Apr11-04, 11:44 AM
<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>\n\n\n\ntessel@um.bot wrote in message news:<c4hcmb\\$vqm\\$1@lfa222122.richmond.edu>...\ n\n> (c) how to obtain and study particular solutions (even for nonlinear PDEs)\n> e.g. using Lie\'s methods, Baecklund symmetries, scattering transform,\n\nIndependent of all else, I\'d be happy to follow your exposition of\nthis topic.\n\nIgor\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>tessel@um.bot wrote in message news:<c4hcmb$vqm$1@lfa222122.richmond.edu>...
> (c) how to obtain and study particular solutions (even for nonlinear PDEs)
> e.g. using Lie's methods, Baecklund symmetries, scattering transform,
Independent of all else, I'd be happy to follow your exposition of
this topic.
Igor
tessel@um.bot
Apr13-04, 05:57 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>Aaron wrote:\n\n> This is generally done in math textbooks,\n\nDo tell! I have easy access to a good math research library, and my\nbackground, as you know, is in math.\n\nI have -never- seen either a math or physics book which discusses the\nissues I listed, so I\'d be very interested if you could (third or fourth\nrequest, arghgh!) provide a -citation- to such a book.\n\n> H. Goldstein _Classical Mechanics_. I don\'t own a copy, so I can\'t tell\n> you precisely what\'s in it, but it\'s one of the standard junior level\n> classical mechanics texts.\n\nThanks, Aaron, this is the book I thought you meant. IIRC it doesn\'t do\nthe trick, and by now I am quite sick of this whole darned subject (I\nknow--- I brought it up, I just didn\'t expect a negative response from you\nin particular), but if I get over this, I\'ll take another look at this\nbook.\n\n> > Or again: when I proposed to try to derive the equation of motion of\n> > "charged" test particles in a given field, did you and Aaron fail to\n> > notice that I was talking about the analogue of the EIH procedure in\n> > the case of gtr? (For other readers: Einstein, Infeld, and Hoffmann\n> > proposed to remove the assumption that the world lines nonspinning\n> > test particles in a vacuum region are timelike geodesics by -deriving-\n> > this from the Einstein field equation.) I assume that you and Aaron\n> > are both aware that this is -highly nontrivial-; if not, see\n>\n> Unless I misunderstand you, this is a standard exercise in many GR\n> books. It\'s done in chapter 9 of Stephani, for example. It\'s not\n> particularly difficult either, so maybe you can elaborate further about\n> what you are referring to?\n\nSorry, I expressed myself very badly. You did misunderstand me. The\ntopic for discussion was -not- how to do EIH for -gtr- but rather how to\ndo something analogous for -other- field theories. This is nontrivial.\nIf you think otherwise, well, by now I am sick of arguing about it.\n\nSeveral other posters suggested further resources--- thanks!\n\nBuzurg Shagird wrote:\n\n> > Since I mentioned those books as inadequate for students like Doug\n> > Sweetser who are trying to formulate and play around with a\n> > "Lagrangian field theory" (even a "toy theory"), you may assume that I\n> > feel they (and the other books I mentioned) do -not- explain -the\n> > "issues for discussion" which I listed- (see below for a repeat of\n> > this list).\n>\n> Well, first of all, when I said that I wasn\'t sure where you see that\n> gap, I did NOT mean ``I know everything there is to know about field\n> theory", NOR ``all possible issues of classical field theory has been\n> discussed in those books." If you got that impression from my post, I am\n> really sorry, that was not my intent at all.\n\nI did not think you were saying that. I thought you were saying you think\nthere are already adequate discussions of the issues I raised suitable for\n-average- students. (I agree that -brilliant- students require much less\nhelp.)\n\n> Your list of ``issues for discussion" includes some things which are\n> covered in the standard textbooks,\n\nCites would be appreciated.\n\n> some things which are covered in the non-standard textbooks, i.e. books\n> which compile existing knowledge, but with a narrow focus or from an\n> uncommon viewpoint,\n\nCites?\n\n> and a few things for which you have look in monographs or research\n> papers.\n\nCites?\n\n> I am unable to see the `gap\' that you mention because I feel that\n> those books prepare you well enough to do your own investigations into\n> the questions that you have asked -- and as far as I could see from\n> your post, you are indeed well prepared to do that investigation.\n\nYes, I agree that these books prepare -me- and they apparently prepared\n-you-, and I have no doubt they will adequately prepare creative and\nhighly motivated students with good background and blessed with\nintelligence, insight, and at least some talent. I am saying that the\nbooks I have seen won\'t prepare students who lack some/all of those\nthings.\n\n> > My claim is quite different: for purposes of playing around\n> > intelligently with Lagrangians, -much more is needed-.\n>\n> Only by those who want to `play around\' with Lagrangians, in the way\n> that you and a few others on this group may want to do. :-) But that is\n> very much a `special interests\' topic. Why would a textbook include all\n> possible ways of playing around with Lagrangians?\n\nWould it help if I dropped the word "textbook"? What if I had proposed a\nbook for graduate students of -philosophy of physics-, rather than\nadvanced undergraduate students of physics?\n\nAnd finally, Doug Sweetser wrote:\n\n> I was in the position four years ago of having field equations, but no\n> Lagrangian. What I did was practice with EM, doing the derivation from\n> the classical EM Lagrange density out to the field equations about a\n> dozen times.\n\nA good way to begin.\n\n> In many ways, it is like practicing scales or multiplication tables:\n> nothing can substitute for practice. After that, it was possible for me\n> to play with variations.\n\nBut the summary you provided for me was so full of misconceptions major\nand minor that I didn\'t even try to list them!\n\n> Before I ramble about anything else, I request you take the above\n> Lagrange density and generate those field equations.\n\nWell, if you were Feynman, you wouldn\'t need to compute anything to see\nwhat will happen here. In a burst of foolish enthusiasm, I promised in\neffect to try to turn everyone into a new Feynman.\n\nSorry, Doug, I can see I am not helping by dropping out here, but I\nalready told you I don\'t have sufficient patience to play by any rules but\nmy own, and by now I am sick of the whole topic and am sorry I ever\nbrought it up. So let\'s just forget it.\n\nThanks to the small number of posters who had a more positive response to\nmy proposed "topics for discussion"! I greatly appreciate the support,\nbut gosh, we were certainly in the minority, arghgh. Alas, I have less\ntime than I anticipated when I offered to lead this discussion. And by\nnow I\'m quite sick of arguing tendentiously about what I regard as\nnonissues, when all I really wanted was to induce people like Aaron to\njust -play-, darn it all.\n\nSo I\'ll withdraw my proposal to discuss these topics.\n\nIgor, in principle I -would- still like to try for something less\nambitious, namely an exposition of "Lie\'s methods, Baecklund symmetries,\nscattering transform". Right now I am too disappointed with the general\ntenor of the response in this thread to contemplate attempting lengthy\nexpository posts in the near future, but time permitting, perhaps I will\ntry, some time later this month [read "millenium"], to summarize the most\nuseful features of Lie\'s theory of symmetries of PDEs, including looking\nfor "particular solutions" and for "Baecklund morphisms".\n\n"T. Essel" (hiding under a stupid psuedonym)\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>Aaron wrote:
> This is generally done in math textbooks,
Do tell! I have easy access to a good math research library, and my
background, as you know, is in math.
I have -never- seen either a math or physics book which discusses the
issues I listed, so I'd be very interested if you could (third or fourth
request, arghgh!) provide a -citation- to such a book.
> H. Goldstein _Classical Mechanics_. I don't own a copy, so I can't tell
> you precisely what's in it, but it's one of the standard junior level
> classical mechanics texts.
Thanks, Aaron, this is the book I thought you meant. IIRC it doesn't do
the trick, and by now I am quite sick of this whole darned subject (I
know--- I brought it up, I just didn't expect a negative response from you
in particular), but if I get over this, I'll take another look at this
book.
> > Or again: when I proposed to try to derive the equation of motion of
> > "charged" test particles in a given field, did you and Aaron fail to
> > notice that I was talking about the analogue of the EIH procedure in
> > the case of gtr? (For other readers: Einstein, Infeld, and Hoffmann
> > proposed to remove the assumption that the world lines nonspinning
> > test particles in a vacuum region are timelike geodesics by -deriving-
> > this from the Einstein field equation.) I assume that you and Aaron
> > are both aware that this is -highly nontrivial-; if not, see
>
> Unless I misunderstand you, this is a standard exercise in many GR
> books. It's done in chapter 9 of Stephani, for example. It's not
> particularly difficult either, so maybe you can elaborate further about
> what you are referring to?
Sorry, I expressed myself very badly. You did misunderstand me. The
topic for discussion was -not- how to do EIH for -gtr- but rather how to
do something analogous for -other- field theories. This is nontrivial.
If you think otherwise, well, by now I am sick of arguing about it.
Several other posters suggested further resources--- thanks!
Buzurg Shagird wrote:
> > Since I mentioned those books as inadequate for students like Doug
> > Sweetser who are trying to formulate and play around with a
> > "Lagrangian field theory" (even a "toy theory"), you may assume that I
> > feel they (and the other books I mentioned) do -not- explain -the
> > "issues for discussion" which I listed- (see below for a repeat of
> > this list).
>
> Well, first of all, when I said that I wasn't sure where you see that
> gap, I did NOT mean ``I know everything there is to know about field
> theory", NOR ``all possible issues of classical field theory has been
> discussed in those books." If you got that impression from my post, I am
> really sorry, that was not my intent at all.
I did not think you were saying that. I thought you were saying you think
there are already adequate discussions of the issues I raised suitable for
-average- students. (I agree that -brilliant- students require much less
help.)
> Your list of ``issues for discussion" includes some things which are
> covered in the standard textbooks,
Cites would be appreciated.
> some things which are covered in the non-standard textbooks, i.e. books
> which compile existing knowledge, but with a narrow focus or from an
> uncommon viewpoint,
Cites?
> and a few things for which you have look in monographs or research
> papers.
Cites?
> I am unable to see the `gap' that you mention because I feel that
> those books prepare you well enough to do your own investigations into
> the questions that you have asked -- and as far as I could see from
> your post, you are indeed well prepared to do that investigation.
Yes, I agree that these books prepare -me- and they apparently prepared
-you-, and I have no doubt they will adequately prepare creative and
highly motivated students with good background and blessed with
intelligence, insight, and at least some talent. I am saying that the
books I have seen won't prepare students who lack some/all of those
things.
> > My claim is quite different: for purposes of playing around
> > intelligently with Lagrangians, -much more is needed-.
>
> Only by those who want to `play around' with Lagrangians, in the way
> that you and a few others on this group may want to do. :-) But that is
> very much a `special interests' topic. Why would a textbook include all
> possible ways of playing around with Lagrangians?
Would it help if I dropped the word "textbook"? What if I had proposed a
book for graduate students of -philosophy of physics-, rather than
advanced undergraduate students of physics?
And finally, Doug Sweetser wrote:
> I was in the position four years ago of having field equations, but no
> Lagrangian. What I did was practice with EM, doing the derivation from
> the classical EM Lagrange density out to the field equations about a
> dozen times.
A good way to begin.
> In many ways, it is like practicing scales or multiplication tables:
> nothing can substitute for practice. After that, it was possible for me
> to play with variations.
But the summary you provided for me was so full of misconceptions major
and minor that I didn't even try to list them!
> Before I ramble about anything else, I request you take the above
> Lagrange density and generate those field equations.
Well, if you were Feynman, you wouldn't need to compute anything to see
what will happen here. In a burst of foolish enthusiasm, I promised in
effect to try to turn everyone into a new Feynman.
Sorry, Doug, I can see I am not helping by dropping out here, but I
already told you I don't have sufficient patience to play by any rules but
my own, and by now I am sick of the whole topic and am sorry I ever
brought it up. So let's just forget it.
Thanks to the small number of posters who had a more positive response to
my proposed "topics for discussion"! I greatly appreciate the support,
but gosh, we were certainly in the minority, arghgh. Alas, I have less
time than I anticipated when I offered to lead this discussion. And by
now I'm quite sick of arguing tendentiously about what I regard as
nonissues, when all I really wanted was to induce people like Aaron to
just -play-, darn it all.
So I'll withdraw my proposal to discuss these topics.
Igor, in principle I -would- still like to try for something less
ambitious, namely an exposition of "Lie's methods, Baecklund symmetries,
scattering transform". Right now I am too disappointed with the general
tenor of the response in this thread to contemplate attempting lengthy
expository posts in the near future, but time permitting, perhaps I will
try, some time later this month [read "millenium"], to summarize the most
useful features of Lie's theory of symmetries of PDEs, including looking
for "particular solutions" and for "Baecklund morphisms".
"T. Essel" (hiding under a stupid psuedonym)
Aaron Bergman
Apr15-04, 11:22 AM
<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>In article <c5hnni\\$ujh\\$1@lfa222122.richmond.edu>, tessel@um.bot wrote:\n\n> Aaron wrote:\n>\n> > This is generally done in math textbooks,\n>\n> Do tell! I have easy access to a good math research library, and my\n> background, as you know, is in math.\n\nI wish I could give you a better answer. Mostly I just remember seeing\nsuch things when browsing in a campus bookstore. Typing \'calculus of\nvariations\' into Amazon returned a number of hits if that helps.\n>\n> I have -never- seen either a math or physics book which discusses the\n> issues I listed, so I\'d be very interested if you could (third or fourth\n> request, arghgh!) provide a -citation- to such a book.\n\nI\'m not sure you\'ll ever find everything you want in a single book,\nespecially at advanced levels. One generally needs to read a number of\nsources to really understand a given subject.\n\n[...]\n\n> Yes, I agree that these books prepare -me- and they apparently prepared\n> -you-, and I have no doubt they will adequately prepare creative and\n> highly motivated students with good background and blessed with\n> intelligence, insight, and at least some talent. I am saying that the\n> books I have seen won\'t prepare students who lack some/all of those\n> things.\n\nThis is probably going to come across poorly, but from my perspective,\nif a student isn\'t willing to go down to the library and do some\nresearch, they probably should consider a different field. Finding a\nsingle source that answers all your questions on any given subject is a\nrare occurrence.\n\nOn the other hand, I think what you say is a pretty neat idea for a\nbook. If it doesn\'t exist, it wouldn\'t be a bad thing if someone wrote\nit.\n\nAs a last thought, another reference that occurred to me in the process\nof writing this is Warren Siegel\'s book "Fields", hep-th/9912205. I\nhaven\'t read it, but I know that it begins with classical field theory\nand then starts quantizing it. You might find it interesting.\n\nAaron\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>In article <c5hnni$ujh$1@lfa222122.richmond.edu>, tessel@um.bot wrote:
> Aaron wrote:
>
> > This is generally done in math textbooks,
>
> Do tell! I have easy access to a good math research library, and my
> background, as you know, is in math.
I wish I could give you a better answer. Mostly I just remember seeing
such things when browsing in a campus bookstore. Typing 'calculus of
variations' into Amazon returned a number of hits if that helps.
>
> I have -never- seen either a math or physics book which discusses the
> issues I listed, so I'd be very interested if you could (third or fourth
> request, arghgh!) provide a -citation- to such a book.
I'm not sure you'll ever find everything you want in a single book,
especially at advanced levels. One generally needs to read a number of
sources to really understand a given subject.
[...]
> Yes, I agree that these books prepare -me- and they apparently prepared
> -you-, and I have no doubt they will adequately prepare creative and
> highly motivated students with good background and blessed with
> intelligence, insight, and at least some talent. I am saying that the
> books I have seen won't prepare students who lack some/all of those
> things.
This is probably going to come across poorly, but from my perspective,
if a student isn't willing to go down to the library and do some
research, they probably should consider a different field. Finding a
single source that answers all your questions on any given subject is a
rare occurrence.
On the other hand, I think what you say is a pretty neat idea for a
book. If it doesn't exist, it wouldn't be a bad thing if someone wrote
it.
As a last thought, another reference that occurred to me in the process
of writing this is Warren Siegel's book "Fields", hep-th/9912205. I
haven't read it, but I know that it begins with classical field theory
and then starts quantizing it. You might find it interesting.
Aaron
Doug Sweetser
Apr15-04, 11:25 AM
<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>Hello:\n\nWhat an odd place to stop a thread. My name appeared several times as\nthe cement-brained Cro-Magnon poster boy (with far gentler comments,\nbut I think that tag is pithier :-) If you\'d rather find the ultimate\nbook, good luck to you, but the citations are not piling up as you\nnoted.\n\nMe, I\'d rather grind. The nice thing about always digging and getting\none\'s hands dirty is that gems can be found on rare occasions. So I\nwas reading a thread (this thread?) recently on SPR, and the author was\nwaxing on how important symmetries in Lagrangians are. Not the first\ntime I have heard this. But I went I looked at the Lagrange density\nagain, this time wanting to dig up a root connected to symmetry. The\nwork took about three minutes tops, located in a different thread.\nThat result may be the key idea driving the proposal because that is\nhow important symmetries in Lagrange densities are.\n\nAs always, personal issues are irrelevant to me. All I care about are\ntechnical critiques of specific equations.\n\ndoug\nquaternions.com\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>Hello:
What an odd place to stop a thread. My name appeared several times as
the cement-brained Cro-Magnon poster boy (with far gentler comments,
but I think that tag is pithier :-) If you'd rather find the ultimate
book, good luck to you, but the citations are not piling up as you
noted.
Me, I'd rather grind. The nice thing about always digging and getting
one's hands dirty is that gems can be found on rare occasions. So I
was reading a thread (this thread?) recently on SPR, and the author was
waxing on how important symmetries in Lagrangians are. Not the first
time I have heard this. But I went I looked at the Lagrange density
again, this time wanting to dig up a root connected to symmetry. The
work took about three minutes tops, located in a different thread.
That result may be the key idea driving the proposal because that is
how important symmetries in Lagrange densities are.
As always, personal issues are irrelevant to me. All I care about are
technical critiques of specific equations.
doug
quaternions.com
Buzurg Shagird
Apr15-04, 11:42 AM
<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>I should probably stay quiet, but ...\n\ntessel@um.bot wrote in message news:<c5hnni\\$ujh\\$1@lfa222122.richmond.edu>...\ n\n> I did not think you were saying that. I thought you were saying you think\n> there are already adequate discussions of the issues I raised suitable for\n> -average- students. (I agree that -brilliant- students require much less\n> help.)\n\nI don\'t know how to make that distinction. I am not being facetious, and I\ndo have some experience with students of physics at all levels. And I mean\nprecisely that I don\'t have a checklist or a rule of thumb that allows me\nto assign labels like that to *students*. To professional researchers, yes,\nbut not to students. I believe that any student, if sufficiently motivated,\ncan learn a subject well enough, from the standard sources, to start doing\noriginal research in it. Some guidance is often helpful, but I think that\nit is needed only to keep up the level of motivation.\n\n> > Your list of ``issues for discussion" includes some things which are\n> > covered in the standard textbooks,\n>\n> Cites would be appreciated.\n\nYou already know which ones I am talking about -- Landau and Lifshitz,\nGoldstein, any introductory book on quantum field theory ... If you want\na page and volume reference, you need to ask a more specific question,\nand hopefully someone who remembers page numbers will be able to help.\n\nOkay, maybe I should give an example.\n\nSay you want an action which produces a linear eq. of motion. Then you need\na Lagrangian quadratic in the fields and their derivatives. Write down all\nsuch terms. That is your Lagrangian.\n\nSuppose you wish to be a bit more serious and want the action to have some\nresemblance to physics. Then (all standard texts) write a term quadratic\nin the time derivative. Add any polynomial in the fields.\n\nSuppose you want the theory to be Lorentz invariant. Then make sure your\nLagrangian is a Lorentz scalar.\n\nSuppose you want the theory to have any other global invariance. Then your\nLagrangian must have that invariance as well.\n\nThat is it, for classical fields.\n\nI have ignored Lagrangians for spinors -- you can construct those in\npretty much the same way.\n\nEverything I said so far is covered in the standard texts. And you know it.\nSo why ask for a cite?\n\n> Would it help if I dropped the word "textbook"? What if I had proposed a\n> book for graduate students of -philosophy of physics-, rather than\n> advanced undergraduate students of physics?\nThen I would suggest starting with a paper by Weinberg, ``What is quantum\nfield theory and what did we think it is?" hep-th/9702027.\nThis paper was presented in a conference on Conceptual foundations of\nquantum field theory, Boston 1996. The proceedings contain several articles\nwhich should be interesting to graduate students of philosophy of physics.\n\n> Well, if you were Feynman, you wouldn\'t need to compute anything to see\n> what will happen here. In a burst of foolish enthusiasm, I promised in\n> effect to try to turn everyone into a new Feynman.\n\nI never met Feynman, but my impression from reading, and talking to, those\nwho had, is that the reason he didn\'t need to compute anything only because\nhe had already spent a lot of time computing everything he wanted to know.\nAnd he could calculate very fast, which helped. :-)\n(Freeman J. Dyson once said that a QED calculation by Feynman was the\nfastest piece of physics calculation he had seen.)\n\n> Sorry, Doug, I can see I am not helping by dropping out here, but I\n> already told you I don\'t have sufficient patience to play by any rules but\n> my own, and by now I am sick of the whole topic and am sorry I ever\n> brought it up. So let\'s just forget it.\n\nWhy not play by your own rules? Giving up is too easy. :-)\n\n-S.\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>I should probably stay quiet, but ...
tessel@um.bot wrote in message news:<c5hnni$ujh$1@lfa222122.richmond.edu>...
> I did not think you were saying that. I thought you were saying you think
> there are already adequate discussions of the issues I raised suitable for
> -average- students. (I agree that -brilliant- students require much less
> help.)
I don't know how to make that distinction. I am not being facetious, and I
do have some experience with students of physics at all levels. And I mean
precisely that I don't have a checklist or a rule of thumb that allows me
to assign labels like that to *students*. To professional researchers, yes,
but not to students. I believe that any student, if sufficiently motivated,
can learn a subject well enough, from the standard sources, to start doing
original research in it. Some guidance is often helpful, but I think that
it is needed only to keep up the level of motivation.
> > Your list of ``issues for discussion" includes some things which are
> > covered in the standard textbooks,
>
> Cites would be appreciated.
You already know which ones I am talking about -- Landau and Lifshitz,
Goldstein, any introductory book on quantum field theory ... If you want
a page and volume reference, you need to ask a more specific question,
and hopefully someone who remembers page numbers will be able to help.
Okay, maybe I should give an example.
Say you want an action which produces a linear eq. of motion. Then you need
a Lagrangian quadratic in the fields and their derivatives. Write down all
such terms. That is your Lagrangian.
Suppose you wish to be a bit more serious and want the action to have some
resemblance to physics. Then (all standard texts) write a term quadratic
in the time derivative. Add any polynomial in the fields.
Suppose you want the theory to be Lorentz invariant. Then make sure your
Lagrangian is a Lorentz scalar.
Suppose you want the theory to have any other global invariance. Then your
Lagrangian must have that invariance as well.
That is it, for classical fields.
I have ignored Lagrangians for spinors -- you can construct those in
pretty much the same way.
Everything I said so far is covered in the standard texts. And you know it.
So why ask for a cite?
> Would it help if I dropped the word "textbook"? What if I had proposed a
> book for graduate students of -philosophy of physics-, rather than
> advanced undergraduate students of physics?
Then I would suggest starting with a paper by Weinberg, ``What is quantum
field theory and what did we think it is?" hep-th/9702027.
This paper was presented in a conference on Conceptual foundations of
quantum field theory, Boston 1996. The proceedings contain several articles
which should be interesting to graduate students of philosophy of physics.
> Well, if you were Feynman, you wouldn't need to compute anything to see
> what will happen here. In a burst of foolish enthusiasm, I promised in
> effect to try to turn everyone into a new Feynman.
I never met Feynman, but my impression from reading, and talking to, those
who had, is that the reason he didn't need to compute anything only because
he had already spent a lot of time computing everything he wanted to know.
And he could calculate very fast, which helped. :-)
(Freeman J. Dyson once said that a QED calculation by Feynman was the
fastest piece of physics calculation he had seen.)
> Sorry, Doug, I can see I am not helping by dropping out here, but I
> already told you I don't have sufficient patience to play by any rules but
> my own, and by now I am sick of the whole topic and am sorry I ever
> brought it up. So let's just forget it.
Why not play by your own rules? Giving up is too easy. :-)
-S.
tessel@tum.bot
Apr19-04, 03:18 AM
<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>On Thu, 15 Apr 2004, Buzurg Shagird wrote:\n\n> I should probably stay quiet, but ...\n\nTo the contrary---- but see my "general comment" below.\n\n> > I did not think you were saying that. I thought you were saying you\n> > think there are already adequate discussions of the issues I raised\n> > suitable for -average- students. (I agree that -brilliant- students\n> > require much less help.)\n>\n> I don\'t know how to make that distinction. I am not being facetious, and\n> I do have some experience with students of physics at all levels. And I\n> mean precisely that I don\'t have a checklist or a rule of thumb that\n> allows me to assign labels like that to *students*. To professional\n> researchers, yes, but not to students. I believe that any student, if\n> sufficiently motivated, can learn a subject well enough, from the\n> standard sources, to start doing original research in it. Some guidance\n> is often helpful, but I think that it is needed only to keep up the\n> level of motivation.\n\nProbably this doesn\'t need saying, but I\'ll say it anyway: the thing which\nmade s.p.r. worthwhile in the past (and might make it worthwhile in the\nfuture) is the presence of\n\n(a) eager, able, and well-motivated students at all levels (possibly\nincluding "amateurs", "continuing education students", and students not\nenrolled in formal courses),\n\n(b) experienced teachers/researchers willing to spend some unremunerated\ntime helping students.\n\nProbably the only thing we disagree about is the degree to which group (a)\nmust be well-prepared by successful solid formal coursework in the\nrequisite math/physics background.\n\nBut your observations suggest a -general comment- concerning discussions\nin s.p.r., at which I hope noone will take offense. I stress that this\ncomment is addressed to -all- current s.p.r. participants, and not to you\nspecifically.\n\nIn my final reply to Doug I pointed out that apparently one thing he and I\nagree upon is that we\'d both like to see more teaching/learning in this\ngroup--- specifically more -technical- discussion of -specific- nontrivial\nissues--- and less -talking- about teaching/learning.\n\nI realize that in this thread I have myself been guilty of -talking- about\nhow I\'d like to do what I\'d like to do, rather than doing it. In fact,\nfeeling myself manuevered into this bad behavior explains why I became\ndespondent! This, together with something beyond my control (lack of\naccess/time/energy), is why I opted out.\n\nBefore anyone gets defensive over my general comment above, I\'d just ask\nthat everyone please remember that in happier days I did follow this\npractice myself. And I hasten to assure newbies that some posters did\nexpress appreciation of my own expository efforts, so such efforts are not\n-entirely- thankless. And we probably agree there can be other rewards\nbesides gratitude: the combination of good teachers and good students\noften results in the -teachers- as well as the students learning something\nvaluable! (John Baez has mentioned this as one motivation for his\nWeeks--- IMO, we would all do well to take this valuable series as a\nmodel.)\n\n> Okay, maybe I should give an example.\n\nHeh, we must both have been thinking the same thing, re my "general\ncomment" :-/\n\n> Say you want an action which produces a linear eq. of motion. Then you need\n> a Lagrangian quadratic in the fields and their derivatives. Write down all\n> such terms. That is your Lagrangian.\n>\n> Suppose you wish to be a bit more serious and want the action to have some\n> resemblance to physics. Then (all standard texts) write a term quadratic\n> in the time derivative. Add any polynomial in the fields.\n\nWell, I did mention sine-Gordon as a valuable example of a naturally\narising equation (naturally arising in pure math, that is, namely\ndifferential geometry!) which -violates- this rule of thumb.\n\nBut of course I agree that mostly one uses polynomials, and in fact I had\nthat in the back of my mind because it implies that whenever we need to\neliminate variables (including derivatives), e.g. by taking further\nderivatives of a given equation in order to eliminate inessential\nparameters, we can avail ourselves of Groebner bases for a suitable\nelimination order. (Actually, trig polynomials are also susceptible to\nthis technique. So are Jacobian elliptic functions, etc.)\n\nI haven\'t yet myself applied elimination orders in quite this way--- at\nleast not in an essential way--- but one thing I was looking forward to\nwas keeping my eye out for opportunities to use this tool (and others) to\ngood effect.\n\n> Suppose you want the theory to be Lorentz invariant. Then make sure your\n> Lagrangian is a Lorentz scalar.\n^^^^^^^^^^^^^^^\n\nA function invariant under the standard action on R^4 of the Lorentz\ngroup?\n\n(For other readers: this would be the notion which is\ngeneralized/systematized by the "point symmetry group" of a [system of]\ndifferential equation[s]. As an application, we can\ngeneralize/systematize the standard Ansatz technique employed in searching\nfor "traveling wave" and "similarity" solutions of PDEs.)\n\n> Suppose you want the theory to have any other global invariance. Then your\n> Lagrangian must have that invariance as well.\n\nAgreed! :-)\n\nWe no doubt agree these particular items are obvious enough (once pointed\nout!), but this excellent list is indeed exactly the kind of thing I was\ntalking about in my proposal for a kind of "Schaum\'s Outline"\nadvice/problem book on field theorizing.\n\n(If you look at past expository threads by me you\'ll see that I tried to\nset exercises which would lead a student to this kind of list of useful\nobservations. Probably we agree about the value of "routine" exercises of\nthis nature.)\n\n> Everything I said so far is covered in the standard texts. And you know it.\n> So why ask for a cite?\n\nSee my "topics for discussion". IMHO, these issues are certainly -not-\n"covered" in the standard texts.\n\n> I never met Feynman, but my impression from reading, and talking to,\n> those who had, is that the reason he didn\'t need to compute anything\n> only because he had already spent a lot of time computing everything he\n> wanted to know.\n\nAnother thing I apparently agree with Doug/you/"everyone" about is the\nvalue of hard slog! Although as you say, for some, like Feynman (or\nWeinberg), slogging is apparently much easier than for most.\n\n> Giving up is too easy. :-)\n\nI -have- been depressed by the way the thread developed, but the real\nobstacle is lack of access/time/energy/(?) civility adequate to the task.\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Thu, 15 Apr 2004, Buzurg Shagird wrote:
> I should probably stay quiet, but ...
To the contrary---- but see my "general comment" below.
> > I did not think you were saying that. I thought you were saying you
> > think there are already adequate discussions of the issues I raised
> > suitable for -average- students. (I agree that -brilliant- students
> > require much less help.)
>
> I don't know how to make that distinction. I am not being facetious, and
> I do have some experience with students of physics at all levels. And I
> mean precisely that I don't have a checklist or a rule of thumb that
> allows me to assign labels like that to *students*. To professional
> researchers, yes, but not to students. I believe that any student, if
> sufficiently motivated, can learn a subject well enough, from the
> standard sources, to start doing original research in it. Some guidance
> is often helpful, but I think that it is needed only to keep up the
> level of motivation.
Probably this doesn't need saying, but I'll say it anyway: the thing which
made s.p.r. worthwhile in the past (and might make it worthwhile in the
future) is the presence of
(a) eager, able, and well-motivated students at all levels (possibly
including "amateurs", "continuing education students", and students not
enrolled in formal courses),
(b) experienced teachers/researchers willing to spend some unremunerated
time helping students.
Probably the only thing we disagree about is the degree to which group (a)
must be well-prepared by successful solid formal coursework in the
requisite math/physics background.
But your observations suggest a -general comment- concerning discussions
in s.p.r., at which I hope noone will take offense. I stress that this
comment is addressed to -all- current s.p.r. participants, and not to you
specifically.
In my final reply to Doug I pointed out that apparently one thing he and I
agree upon is that we'd both like to see more teaching/learning in this
group--- specifically more -technical- discussion of -specific- nontrivial
issues--- and less -talking- about teaching/learning.
I realize that in this thread I have myself been guilty of -talking- about
how I'd like to do what I'd like to do, rather than doing it. In fact,
feeling myself manuevered into this bad behavior explains why I became
despondent! This, together with something beyond my control (lack of
access/time/energy), is why I opted out.
Before anyone gets defensive over my general comment above, I'd just ask
that everyone please remember that in happier days I did follow this
practice myself. And I hasten to assure newbies that some posters did
express appreciation of my own expository efforts, so such efforts are not
-entirely- thankless. And we probably agree there can be other rewards
besides gratitude: the combination of good teachers and good students
often results in the -teachers- as well as the students learning something
valuable! (John Baez has mentioned this as one motivation for his
Weeks--- IMO, we would all do well to take this valuable series as a
model.)
> Okay, maybe I should give an example.
Heh, we must both have been thinking the same thing, re my "general
comment" :-/
> Say you want an action which produces a linear eq. of motion. Then you need
> a Lagrangian quadratic in the fields and their derivatives. Write down all
> such terms. That is your Lagrangian.
>
> Suppose you wish to be a bit more serious and want the action to have some
> resemblance to physics. Then (all standard texts) write a term quadratic
> in the time derivative. Add any polynomial in the fields.
Well, I did mention sine-Gordon as a valuable example of a naturally
arising equation (naturally arising in pure math, that is, namely
differential geometry!) which -violates- this rule of thumb.
But of course I agree that mostly one uses polynomials, and in fact I had
that in the back of my mind because it implies that whenever we need to
eliminate variables (including derivatives), e.g. by taking further
derivatives of a given equation in order to eliminate inessential
parameters, we can avail ourselves of Groebner bases for a suitable
elimination order. (Actually, trig polynomials are also susceptible to
this technique. So are Jacobian elliptic functions, etc.)
I haven't yet myself applied elimination orders in quite this way--- at
least not in an essential way--- but one thing I was looking forward to
was keeping my eye out for opportunities to use this tool (and others) to
good effect.
> Suppose you want the theory to be Lorentz invariant. Then make sure your
> Lagrangian is a Lorentz scalar.
^^^^^^^^^^^^^^^
A function invariant under the standard action on R^4 of the Lorentz
group?
(For other readers: this would be the notion which is
generalized/systematized by the "point symmetry group" of a [system of]
differential equation[s]. As an application, we can
generalize/systematize the standard Ansatz technique employed in searching
for "traveling wave" and "similarity" solutions of PDEs.)
> Suppose you want the theory to have any other global invariance. Then your
> Lagrangian must have that invariance as well.
Agreed! :-)
We no doubt agree these particular items are obvious enough (once pointed
out!), but this excellent list is indeed exactly the kind of thing I was
talking about in my proposal for a kind of "Schaum's Outline"
advice/problem book on field theorizing.
(If you look at past expository threads by me you'll see that I tried to
set exercises which would lead a student to this kind of list of useful
observations. Probably we agree about the value of "routine" exercises of
this nature.)
> Everything I said so far is covered in the standard texts. And you know it.
> So why ask for a cite?
See my "topics for discussion". IMHO, these issues are certainly -not-
"covered" in the standard texts.
> I never met Feynman, but my impression from reading, and talking to,
> those who had, is that the reason he didn't need to compute anything
> only because he had already spent a lot of time computing everything he
> wanted to know.
Another thing I apparently agree with Doug/you/"everyone" about is the
value of hard slog! Although as you say, for some, like Feynman (or
Weinberg), slogging is apparently much easier than for most.
> Giving up is too easy. :-)
I -have- been depressed by the way the thread developed, but the real
obstacle is lack of access/time/energy/(?) civility adequate to the task.
"T. Essel" (hiding somewhere in cyberspace)
tessel@tum.bot
Apr19-04, 02:21 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>On Thu, 15 Apr 2004, Doug Sweetser wrote:\n\n> What an odd place to stop a thread. My name appeared several times as\n> the cement-brained Cro-Magnon poster boy (with far gentler comments, but\n> I think that tag is pithier :-)\n>\n> Me, I\'d rather grind. The nice thing about always digging and getting\n> one\'s hands dirty is that gems can be found on rare occasions.\n\nAlthough I did repeatedly cite you as an exemplar of a seriously confused\nstudent, you certainly are not the only one I had in mind! I just\ncouldn\'t remember anyone else\'s name off the top of my head.\n\nAnd if I really thought you were "cement-brained", of course I wouldn\'t\nhave tried to engage you as a "Lagrangian playmate"! But in any case, as\nit turns out--- as I said a few weeks ago--- I don\'t after all have\nsufficient access/energy/time/civility(?) to pursue the ambitious thread I\nimpulsively tried--- with good intentions--- to initiate.\n\nBecause my intention was certainly not to discourage you from playing, I\nwill attempt one last time to clarify what I -was- trying to urge you to\ndo.\n\nFirst, let me say that I -entirely agree- with you concerning the value of\nplaying with specific Lagrangians! And of course I entirely approve of\nthe search for "gems"!\n\n> As always, personal issues are irrelevant to me. All I care about are\n> technical critiques of specific equations.\n\nActually, I -do- want you (and me, and other physics fans) to -continue-\nto "get your hands dirty" by playing with specific models. My suggestion\nwas certainly not to stop playing!\n\nRather, I said that I think that you are in over your head (wrt your\ncurrent level of insight/sophistication) regarding -the specific\nLagrangian- you have been trying to study. I suggested that you back off\nfrom that -specific- proposal for the time being. I proposed that instead\nyou\n\n> start as simply as possible and slowly increase the level of\n> sophistication,\n\nstudying\n\n> (not neccessarily in this order)\n>\n> 1. first linear, then nonlinear Lagrangians,\n>\n> 2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more\n> variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),\n>\n> 3. first scalar theories, then vector theories, then tensor theories,\n>\n> 4. first classical field theories, then quantum field theories,\n\nI suggested that for various specific Lagrangians you systematically\nexplore, with our help, the following issues (this was not neccessarily an\nexhaustive list!):\n\n> 1. how to sensibly select the terms of our Lagrangian, as written\n\n[when appropriate]\n\n> in the useful form\n>\n> L = L_matt + L_interaction + L_field\n>\n> 2. how to guess key features of the "free-field" case of the field\n> equation of our theory,\n>\n> 3. how to write down the Euler-Lagrange equations (allowing for the\n> possibility of second and higher order derivatives),\n>\n> 4. how to verify key features of the (free-field) field equation, e.g.\n>\n> (a) how to determine the group of "point symmetries" and the subgroup of\n> "variational symmetries",\n>\n> (b) how to confirm the existence of wavelike solutions to the\n> (free-field) field equation,\n>\n> (c) how to obtain and study particular solutions (even for nonlinear\n> PDEs) e.g. using Lie\'s methods, Baecklund symmetries, scattering\n> transform,\n>\n> 5. how to obtain a suitable energy-momentum tensor (and what to do if\n> the "canonical energy-momentum tensor" turns out to be -nonsymmetric-),\n>\n> 6. more generally how to find "conserved currents" (e.g. from\n> "divergence symmetries"), at least in the case where L only contains\n> first order derivatives,\n>\n> 7. how to find suitable "conserved currents" if L contains second order\n> derivatives,\n>\n> 8. how to determine L_interaction from energy-momentum exchange between\n> the field and "particles", if this is possible, or if not, how otherwise\n> to obtain a suitable "equation of motion" for "charged" particles,\n>\n> 9. how to check for at least the most commonly encountered\n> self-inconsistencies which can arise in a field theory,\n\nI suggested that this kind of systematic exploration would help you/others\nto develop a -highly valuable skill-: the ability to write down an\nintelligently chosen Lagrangian leading to a field theory which has a\n"reasonable chance" of achieving a stated goal (e.g. behaving like\nMaxwell\'s theory under common laboratory conditions, while achieving\nsomething Maxwell\'s theory does not, such as ameliorating point charge\nsingularities).\n\nIf you followed this advice, I am confident that you would also acquire\nthe ability to quickly see, in many cases, what is wrong with an\n"obviously bad" choice of Lagrangian--- such as the one you proposed.\nYou would also be more likely, I think, to avoid wasting the time/energy\nof the amazingly generous email correspondent you mentioned and of various\nposters here. Certainly you would be much more likely to obtain here the\nkind of -focused- feedback which -immediately improves your understanding-\nof some specific issue.\n\n(The "wastage" which I and others have cited is clearly due to multiple\nmisunderstandings concerning numerous issues at various levels of\nsophistication. This kind of mutual multiple misunderstanding is -much-\nmore likely to arise if any participant is in over his head, and once it\nhas happened, IMO, the only real fix is to erase the whole discussion and\nto start all over again at a lower level.)\n\nAnd, while I have no idea what would turn up if you followed my advice, in\nmy experience systematic exploration along the lines I suggested is the\nbest way to uncover "gems". To be sure, I would expect that any "gems"\nuncovered might not be directly relevant to the original goal (developing\nthe skill of intelligent field theorizing), but might involve some\ncompletely unexpected insight into who-knows-what. But that\'s probably\nhalf the fun of finding a mathematical gem--- typically, the lucky\nexplorer finds not just one stone but a veritable Kimberly. And to keep\nup the metaphor, the stones aren\'t diamonds or rubies but something\n-completely new-, yet of comparable beauty.\n\nSo I also regret that this thread never developed (as was my goal) into\nthe specific critique of specific proposed Lagrangians. And I vigorously\napplaud your desire for this kind of focused discussion on technical\nissues! I think our only disagreement was over -which- Lagrangians to\ndiscuss -when-. And if you reconsider following my advice, others here\nare certainly capable of helping you even if I cannot after all\nparticipate myself.\n\n(I haven\'t -completely- wimped out--- I do still hope to exposit here,\nwhen I have more access/time/energy, the computation/applications of\nsymmetry groups of partial differential equations and perhaps additional\ntechniques such as scattering theory and Baecklund morphisms.)\n\nWishing you the best of luck with your future Lagrangian play,\n\n"T. Essel" (hiding somewhere in cyberspace)\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>On Thu, 15 Apr 2004, Doug Sweetser wrote:
> What an odd place to stop a thread. My name appeared several times as
> the cement-brained Cro-Magnon poster boy (with far gentler comments, but
> I think that tag is pithier :-)
>
> Me, I'd rather grind. The nice thing about always digging and getting
> one's hands dirty is that gems can be found on rare occasions.
Although I did repeatedly cite you as an exemplar of a seriously confused
student, you certainly are not the only one I had in mind! I just
couldn't remember anyone else's name off the top of my head.
And if I really thought you were "cement-brained", of course I wouldn't
have tried to engage you as a "Lagrangian playmate"! But in any case, as
it turns out--- as I said a few weeks ago--- I don't after all have
sufficient access/energy/time/civility(?) to pursue the ambitious thread I
impulsively tried--- with good intentions--- to initiate.
Because my intention was certainly not to discourage you from playing, I
will attempt one last time to clarify what I -was- trying to urge you to
do.
First, let me say that I -entirely agree- with you concerning the value of
playing with specific Lagrangians! And of course I entirely approve of
the search for "gems"!
> As always, personal issues are irrelevant to me. All I care about are
> technical critiques of specific equations.
Actually, I -do- want you (and me, and other physics fans) to -continue-
to "get your hands dirty" by playing with specific models. My suggestion
was certainly not to stop playing!
Rather, I said that I think that you are in over your head (wrt your
current level of insight/sophistication) regarding -the specific
Lagrangian- you have been trying to study. I suggested that you back off
from that -specific- proposal for the time being. I proposed that instead
you
> start as simply as possible and slowly increase the level of
> sophistication,
studying
> (not neccessarily in this order)
>
> 1. first linear, then nonlinear Lagrangians,
>
> 2. first Lagrangians of form L(x,u,u_x), then L(x,t,u,u_x,u_t) (more
> variables!) and L(x,u,u_x, u_(xx)) (higher order derivatives!),
>
> 3. first scalar theories, then vector theories, then tensor theories,
>
> 4. first classical field theories, then quantum field theories,
I suggested that for various specific Lagrangians you systematically
explore, with our help, the following issues (this was not neccessarily an
exhaustive list!):
> 1. how to sensibly select the terms of our Lagrangian, as written
[when appropriate]
> in the useful form
>
> L = L_{matt} + L_{interaction} + L_{field}
>
> 2. how to guess key features of the "free-field" case of the field
> equation of our theory,
>
> 3. how to write down the Euler-Lagrange equations (allowing for the
> possibility of second and higher order derivatives),
>
> 4. how to verify key features of the (free-field) field equation, e.g.
>
> (a) how to determine the group of "point symmetries" and the subgroup of
> "variational symmetries",
>
> (b) how to confirm the existence of wavelike solutions to the
> (free-field) field equation,
>
> (c) how to obtain and study particular solutions (even for nonlinear
> PDEs) e.g. using Lie's methods, Baecklund symmetries, scattering
> transform,
>
> 5. how to obtain a suitable energy-momentum tensor (and what to do if
> the "canonical energy-momentum tensor" turns out to be -nonsymmetric-),
>
> 6. more generally how to find "conserved currents" (e.g. from
> "divergence symmetries"), at least in the case where L only contains
> first order derivatives,
>
> 7. how to find suitable "conserved currents" if L contains second order
> derivatives,
>
> 8. how to determine L_{interaction} from energy-momentum exchange between
> the field and "particles", if this is possible, or if not, how otherwise
> to obtain a suitable "equation of motion" for "charged" particles,
>
> 9. how to check for at least the most commonly encountered
> self-inconsistencies which can arise in a field theory,
I suggested that this kind of systematic exploration would help you/others
to develop a -highly valuable skill-: the ability to write down an
intelligently chosen Lagrangian leading to a field theory which has a
"reasonable chance" of achieving a stated goal (e.g. behaving like
Maxwell's theory under common laboratory conditions, while achieving
something Maxwell's theory does not, such as ameliorating point charge
singularities).
If you followed this advice, I am confident that you would also acquire
the ability to quickly see, in many cases, what is wrong with an
"obviously bad" choice of Lagrangian--- such as the one you proposed.
You would also be more likely, I think, to avoid wasting the time/energy
of the amazingly generous email correspondent you mentioned and of various
posters here. Certainly you would be much more likely to obtain here the
kind of -focused- feedback which -immediately improves your understanding-
of some specific issue.
(The "wastage" which I and others have cited is clearly due to multiple
misunderstandings concerning numerous issues at various levels of
sophistication. This kind of mutual multiple misunderstanding is -much-
more likely to arise if any participant is in over his head, and once it
has happened, IMO, the only real fix is to erase the whole discussion and
to start all over again at a lower level.)
And, while I have no idea what would turn up if you followed my advice, in
my experience systematic exploration along the lines I suggested is the
best way to uncover "gems". To be sure, I would expect that any "gems"
uncovered might not be directly relevant to the original goal (developing
the skill of intelligent field theorizing), but might involve some
completely unexpected insight into who-knows-what. But that's probably
half the fun of finding a mathematical gem--- typically, the lucky
explorer finds not just one stone but a veritable Kimberly. And to keep
up the metaphor, the stones aren't diamonds or rubies but something
-completely new-, yet of comparable beauty.
So I also regret that this thread never developed (as was my goal) into
the specific critique of specific proposed Lagrangians. And I vigorously
applaud your desire for this kind of focused discussion on technical
issues! I think our only disagreement was over -which- Lagrangians to
discuss -when-. And if you reconsider following my advice, others here
are certainly capable of helping you even if I cannot after all
participate myself.
(I haven't -completely- wimped out--- I do still hope to exposit here,
when I have more access/time/energy, the computation/applications of
symmetry groups of partial differential equations and perhaps additional
techniques such as scattering theory and Baecklund morphisms.)
Wishing you the best of luck with your future Lagrangian play,
"T. Essel" (hiding somewhere in cyberspace)
Doug Sweetser
Apr22-04, 04:04 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>Hello T Essel:\n\nLet me respond to one point of your post based on my experiences.\n\n> In my final reply to Doug I pointed out that apparently one thing he\n> and I agree upon is that we\'d both like to see more teaching/learning\n> in this group--- specifically more -technical- discussion of\n> -specific- nontrivial issues--- and less -talking- about\n> teaching/learning.\n\nThis is difficult to do either in a newsgroup or in a technical book.\nThe reason is typography. With a pencil, it is easy to write a great\nvariety of symbols. With ASCII, it is a greater challenge. Even with\nLaTeX, people write far fewer equations than are down with pencil.\n\nI have done ASCII derivations to this newsgroup. It is my observation\nthat few reply to such posts. The monospaced letters don\'t translate\nefficiently to how math by pencils look, so most hit a few rapid\nspacebars and move on. I do that myself!\n\nFor a series of talks I gave, I decided to put in the effort of\nrecreating with LaTeX what I would do with a pencil. That required a\nBIG block of time (I was unemployed at the time). I used texmacs which\nwas designed to speed things up. I had a policy of always stating\nexactly where I was starting from. Usually in under 8 steps I got to\nthe end. No steps were "left for the reader" :-) To make it available\nin both pdf and html also was a time sink. I enjoyed this time\nunemployed.\n\nHow things look matters to communication efficiency,\ndoug\nquaternions.com\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>Hello T Essel:
Let me respond to one point of your post based on my experiences.
> In my final reply to Doug I pointed out that apparently one thing he
> and I agree upon is that we'd both like to see more teaching/learning
> in this group--- specifically more -technical- discussion of
> -specific- nontrivial issues--- and less -talking- about
> teaching/learning.
This is difficult to do either in a newsgroup or in a technical book.
The reason is typography. With a pencil, it is easy to write a great
variety of symbols. With ASCII, it is a greater challenge. Even with
LaTeX, people write far fewer equations than are down with pencil.
I have done ASCII derivations to this newsgroup. It is my observation
that few reply to such posts. The monospaced letters don't translate
efficiently to how math by pencils look, so most hit a few rapid
spacebars and move on. I do that myself!
For a series of talks I gave, I decided to put in the effort of
recreating with LaTeX what I would do with a pencil. That required a
BIG block of time (I was unemployed at the time). I used texmacs which
was designed to speed things up. I had a policy of always stating
exactly where I was starting from. Usually in under 8 steps I got to
the end. No steps were "left for the reader" :-) To make it available
in both pdf and html also was a time sink. I enjoyed this time
unemployed.
How things look matters to communication efficiency,
doug
quaternions.com
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