Field theory for continua [Archive]

Van Jacques

Oct22-04, 12:20 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>\n\nThe following is on "classical" theory--no quantum mechanics, though\nits relativistic.\nI have written a paper on this, which I would like to upload to\nthe archive, but have no-one to vouch for it. If I am mistaken and its\nnot worth archiving, I would like to know, so I can forget about this.\n===========\nI recall Goldstein\'s "Classical Mechanics" does Lagrangian and\nHamiltonian mechanics for particles, then tries but fails to do\ncontinua in the last chapter--a rather pathetic finish to an\notherwise great text. One can now\ninclude continua and use differential geometry.\nI have posted this before to a deafening silence.\nThe mechanics of continua is not a sexy topic, but it is basic,\nlike EM and particle motion.\n\nWhen I learned EM as a classical field theory from L&L , Jackson,\nand Misner,Thorne and Wheeler (MTW), I was impressed, as\nI think everyone is.\n\ndF = 0 ==> F = dA, L = - (F|F)/2 + (J|A) ==>\n\nd*F = 0 and T = gL - dA \\x dL/D(dA) = canonical EMT for the EM field.\n\nBut what about continuous matter--fluids, plasma, MHD, etc.?\nHas this appeared in all the texts while I haven\'t been looking?\nIf so, can someone give me a ref.?\nI have been out of touch for some years, have things changed?\nIsn\'t field theory--the fields like A or z^i and the Lagrangians of any\nimportance any more? I thought they were basic. This is the best,\nand really the only way for advanced work on motion, waves, and\nany further study of fluids, plasma, and MHD--all continua.\nI know everyone is busy with their own work, but I would be grateful\nfor any response to the following.\n\nDoes everyone now know how to do continua as a field theory?\n\nd*J = 0 ==> *J = n*u = dz^1 /\\ dz^2 /\\ dz^3 ;\n\nso the eqn. of continuity implies the existence of the 3 scalar fields\nz^i for continua?\n\nThen classical field theory gives the EMT and eqns. of motion:\n\nT = gL - dz^i \\x dL/dz^i ; where for cold matter L = - n = sqrt(*J|*J).\n\n(The above is valid in GR, but its best to do it for SR first).\nDoes anyone have any comments on this. The same can be done\nfor MHD, plasma, or any matter that obeys the conservation of mass\nd*J = 0.\n\nIs it too basic too talk about? Is this well known now. It wasn\'t\nknown at all when I learned physics, and I have seen no changes since\nthen, except advances in particle theory.\n\nIs it thought that matter in space, which makes up our universe, the\ngalaxies, etc. is not important enough to talk about?\n\nI would appreciate some comment from anyone.\n\nWhat have I missed? Where have I gone wrong in thinking this is worth\nputting in the texts?\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>The following is on "classical" theory--no quantum mechanics, though
its relativistic.
I have written a paper on this, which I would like to upload to
the archive, but have no-one to vouch for it. If I am mistaken and its
not worth archiving, I would like to know, so I can forget about this.
===========
I recall Goldstein's "Classical Mechanics" does Lagrangian and
Hamiltonian mechanics for particles, then tries but fails to do
continua in the last chapter--a rather pathetic finish to an
otherwise great text. One can now
include continua and use differential geometry.
I have posted this before to a deafening silence.
The mechanics of continua is not a sexy topic, but it is basic,
like EM and particle motion.

When I learned EM as a classical field theory from L&L , Jackson,
and Misner,Thorne and Wheeler (MTW), I was impressed, as
I think everyone is.

dF =[/itex] ==> F = dA, L = - (F|F)/2 + (J|A) ==>

d*F = and T = gL - dA \x dL/D(dA) = canonical EMT for the EM field.

But what about continuous matter--fluids, plasma, MHD, etc.?
Has this appeared in all the texts while I haven't been looking?
If so, can someone give me a ref.?
I have been out of touch for some years, have things changed?
Isn't field theory--the fields like A or z^i and the Lagrangians of any
importance any more? I thought they were basic. This is the best,
and really the only way for advanced work on motion, waves, and
any further study of fluids, plasma, and MHD--all continua.
I know everyone is busy with their own work, but I would be grateful
for any response to the following.

Does everyone now know how to do continua as a field theory?

d*J = ==> [itex]*J = n*u = dz^1 /\ dz^2 /\ dz^3 ;

so the eqn. of continuity implies the existence of the 3 scalar fields
z^i for continua?

Then classical field theory gives the EMT and eqns. of motion:

T = gL - dz^i \x dL/dz^i ; where for cold matter L = - n = \sqrt(*J|*J).

(The above is valid in GR, but its best to do it for SR first).
Does anyone have any comments on this. The same can be done
for MHD, plasma, or any matter that obeys the conservation of mass
d*J = .

Is it too basic too talk about? Is this well known now. It wasn't
known at all when I learned physics, and I have seen no changes since
then, except advances in particle theory.

Is it thought that matter in space, which makes up our universe, the
galaxies, etc. is not important enough to talk about?

I would appreciate some comment from anyone.

What have I missed? Where have I gone wrong in thinking this is worth
putting in the texts?

Igor Khavkine

Oct24-04, 09:04 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>\nOn 2004-10-22, Van Jacques <vanjac12@yahoo.com> wrote:\n\n> I recall Goldstein\'s "Classical Mechanics" does Lagrangian and\n> Hamiltonian mechanics for particles, then tries but fails to do\n> continua in the last chapter--a rather pathetic finish to an\n> otherwise great text. One can now\n> include continua and use differential geometry.\n\nI\'m not sure how Goldstein "fails to do continua" in the last chapter.\nI don\'t believe the treatment is erroneous, but it is brief compared to\nthe amount of space devoted to other topics. I\'ve found that the use of\ndifferential geometry in physics is dictated more by the culture of a\ngiven subfield instead of whatever is fashionable in modern mathematics.\nContinuum mechanics can be done quite well with the more orthodox use of\nvector analysis. In fact, this is what is done in most texts on fluid\ndynamics and elasticity theory (for example the classic texts by Landau\nand Lifshitz).\n\n> I have posted this before to a deafening silence.\n\nUnfortunately, sometimes questions like "What do you think?"\nare answered implicitly in this way.\n\n> The mechanics of continua is not a sexy topic, but it is basic,\n> like EM and particle motion.\n\nI would agree. However, it seems that over the past few decades\ncontinuum mechanics has left the regular curriculum of undergraduate\nphysics. This subject seems to have moved over to mechanical\nengineering. And because people tend to use what they know, continuum\nmechanics seems to have lost influence over the thoughts of those\nnow engaged in research.\n\n> When I learned EM as a classical field theory from L&L , Jackson,\n> and Misner,Thorne and Wheeler (MTW), I was impressed, as\n> I think everyone is.\n>\n> dF = 0 ==> F = dA, L = - (F|F)/2 + (J|A) ==>\n>\n> d*F = 0 and T = gL - dA \\x dL/D(dA) = canonical EMT for the EM field.\n>\n> But what about continuous matter--fluids, plasma, MHD, etc.?\n> Has this appeared in all the texts while I haven\'t been looking?\n> If so, can someone give me a ref.?\n> I have been out of touch for some years, have things changed?\n> Isn\'t field theory--the fields like A or z^i and the Lagrangians of any\n> importance any more? I thought they were basic. This is the best,\n> and really the only way for advanced work on motion, waves, and\n> any further study of fluids, plasma, and MHD--all continua.\n> I know everyone is busy with their own work, but I would be grateful\n> for any response to the following.\n\nI know little about these topics, but from what I\'ve seen the research\non these topics is still active but segregated into its own community.\nThat would explain why one does not hear about it very often.\n\n> Does everyone now know how to do continua as a field theory?\n>\n> d*J = 0 ==> *J = n*u = dz^1 /\\ dz^2 /\\ dz^3 ;\n>\n> so the eqn. of continuity implies the existence of the 3 scalar fields\n> z^i for continua?\n>\n> Then classical field theory gives the EMT and eqns. of motion:\n>\n> T = gL - dz^i \\x dL/dz^i ; where for cold matter L = - n = sqrt(*J|*J).\n>\n> (The above is valid in GR, but its best to do it for SR first).\n> Does anyone have any comments on this. The same can be done\n> for MHD, plasma, or any matter that obeys the conservation of mass\n> d*J = 0.\n>\n> Is it too basic too talk about? Is this well known now. It wasn\'t\n> known at all when I learned physics, and I have seen no changes since\n> then, except advances in particle theory.\n\nAlthough I still find your equations a little hard to decipher, I think\nall of this is well known by now, even if it is usually expressed in\ndifferent language. Even if there has been some advance in this field,\nit would hardly have made it to the mainstream presses. After all,\nunderlying this theory are just basic Newton\'s laws of motion.\n\nIf you are interested in how fluid dynamics is done in a relativistic\nsetting, I think astrophysics is the right place to look for the\nliterature. Otherwise, most fluid dymaics work that I\'ve seen is quite\nnon-relativistic.\n\n> I would appreciate some comment from anyone.\n\nI\'ve learned that formulating a theory is rarely enough to attract\nattention. The theory must be applied to solve some problems. I don\'t\nknow if there would be an audience for your paper. But if you wish to\ntalk about it, you could post here some worked out problems using this\ntheory. I\'m sure it would be enlightening and a nice exercise in\ndifferential geometry and continuum mechanics for anyone who follows.\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>On 2004-10-22, Van Jacques <vanjac12@yahoo.com> wrote:

> I recall Goldstein's "Classical Mechanics" does Lagrangian and
> Hamiltonian mechanics for particles, then tries but fails to do
> continua in the last chapter--a rather pathetic finish to an
> otherwise great text. One can now
> include continua and use differential geometry.

I'm not sure how Goldstein "fails to do continua" in the last chapter.
I don't believe the treatment is erroneous, but it is brief compared to
the amount of space devoted to other topics. I've found that the use of
differential geometry in physics is dictated more by the culture of a
given subfield instead of whatever is fashionable in modern mathematics.
Continuum mechanics can be done quite well with the more orthodox use of
vector analysis. In fact, this is what is done in most texts on fluid
dynamics and elasticity theory (for example the classic texts by Landau
and Lifshitz).

> I have posted this before to a deafening silence.

Unfortunately, sometimes questions like "What do you think?"
are answered implicitly in this way.

> The mechanics of continua is not a sexy topic, but it is basic,
> like EM and particle motion.

I would agree. However, it seems that over the past few decades
continuum mechanics has left the regular curriculum of undergraduate
physics. This subject seems to have moved over to mechanical
engineering. And because people tend to use what they know, continuum
mechanics seems to have lost influence over the thoughts of those
now engaged in research.

> When I learned EM as a classical field theory from L&L , Jackson,
> and Misner,Thorne and Wheeler (MTW), I was impressed, as
> I think everyone is.
>
> dF = ==> F = dA, L = - (F|F)/2 + (J|A) ==>
>
> d*F = and T = gL - dA \x dL/D(dA) = canonical EMT for the EM field.
>
> But what about continuous matter--fluids, plasma, MHD, etc.?
> Has this appeared in all the texts while I haven't been looking?
> If so, can someone give me a ref.?
> I have been out of touch for some years, have things changed?
> Isn't field theory--the fields like A or z^i and the Lagrangians of any
> importance any more? I thought they were basic. This is the best,
> and really the only way for advanced work on motion, waves, and
> any further study of fluids, plasma, and MHD--all continua.
> I know everyone is busy with their own work, but I would be grateful
> for any response to the following.

I know little about these topics, but from what I've seen the research
on these topics is still active but segregated into its own community.
That would explain why one does not hear about it very often.

> Does everyone now know how to do continua as a field theory?
>
> d*J = ==> *J = n*u = dz^1 /\ dz^2 /\ dz^3 ;
>
> so the eqn. of continuity implies the existence of the 3 scalar fields
> z^i for continua?
>
> Then classical field theory gives the EMT and eqns. of motion:
>
> T = gL - dz^i \x dL/dz^i ; where for cold matter L = - n = \sqrt(*J|*J).
>
> (The above is valid in GR, but its best to do it for SR first).
> Does anyone have any comments on this. The same can be done
> for MHD, plasma, or any matter that obeys the conservation of mass
> d*J = .
>
> Is it too basic too talk about? Is this well known now. It wasn't
> known at all when I learned physics, and I have seen no changes since
> then, except advances in particle theory.

Although I still find your equations a little hard to decipher, I think
all of this is well known by now, even if it is usually expressed in
different language. Even if there has been some advance in this field,
it would hardly have made it to the mainstream presses. After all,
underlying this theory are just basic Newton's laws of motion.

If you are interested in how fluid dynamics is done in a relativistic
setting, I think astrophysics is the right place to look for the
literature. Otherwise, most fluid dymaics work that I've seen is quite
non-relativistic.

> I would appreciate some comment from anyone.

I've learned that formulating a theory is rarely enough to attract
attention. The theory must be applied to solve some problems. I don't
know if there would be an audience for your paper. But if you wish to
talk about it, you could post here some worked out problems using this
theory. I'm sure it would be enlightening and a nice exercise in
differential geometry and continuum mechanics for anyone who follows.

Igor

Arnold Neumaier

Oct24-04, 09:04 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>\nVan Jacques wrote:\n\n> But what about continuous matter--fluids, plasma, MHD, etc.?\n> Has this appeared in all the texts while I haven\'t been looking?\n> If so, can someone give me a ref.?\n>\n> Is it thought that matter in space, which makes up our universe, the\n> galaxies, etc. is not important enough to talk about?\n\nTry\n\nP. J. Morrison,\nHamiltonian description of the ideal fluid,\nRev. Mod. Phys., 70, 467--521 (1998).\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>Van Jacques wrote:

> But what about continuous matter--fluids, plasma, MHD, etc.?
> Has this appeared in all the texts while I haven't been looking?
> If so, can someone give me a ref.?
>
> Is it thought that matter in space, which makes up our universe, the
> galaxies, etc. is not important enough to talk about?

Try

P. J. Morrison,
Hamiltonian description of the ideal fluid,
Rev. Mod. Phys., 70, 467--521 (1998).

Van Jacques

Oct25-04, 08:09 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\nWhat I am saying in my post is this. A physical theory is based on\nHamilton\'s principle of least action and canonical field theory. The\nproblem is\n\n1) what is the field?\n2) What is the Lagragian?\n\nThen one just cranks the machinery of classical field theory to get\nthe Euler Lagrange eqn. and/or the energy momentum tensor (the EMT) T.\n\nFirst consider EM because people are (strangely) more familiar with EM\nthan with the theory for (continuous) matter. The basic homogeneous\neqns. for the EM field 2-form F are dF = 0, which give F = dA,\nwhere A is the vector field (actually the 1-form) for the field theory\nof EM. We say that F is closed and therefore exact.\nThere is no guessing or confusion about it, we know how to do this.\nWe also know that the Lagrangian is\n\nL = - (F|F)/2 + (K|A) ;\n\nHamilton\'s principle and canonical field theory then tell us everything\nabout the EM field; the inhomogeneous eqn. d*F = K\n= electric current density, (* = Hodge operator) and the EMT\n\nT = gL - dA \\x DL/D(dA) (see any text, \\x = direct product,\n\nD = gradient operator or covariant derivative,\nd = exterior derivative.\n\nThis should all be familiar to anyone who has done fields in Landau\nand Lifshitz or Jackson.\n\nNow the point of the post. What about continua. What is the\nhomogenous eqn. which gives the field for continua?\nIt is the eqn. of continuity. Let j = *J = n*u, where\nJ = nu = matter current density, n = (rest) mass density,\nand u = 4-velocity. Then write the eqn. of continuity div(J) = 0\nas dj = 0. The exterior derivative of the 3-form j = 0,\nso that j is closed and therefore exact. Because of the isotropy\nof 3D space, we have j is the exterior product of the exterior\nderivative of 3 scalar fields z^1, z^2, z^3.\n\nj = dz^1 /\\ dz^2 /\\ dz^3 = *J\n\nso that dj = 0 = *[div(J)] and the eqn. of continuity or conservation\nof mass is automatically satified.\n\nThis becomes even more clear for motion in 1 spatial dimension,\nor 2D spacetime. Then both J and j = *J are 1-forms, and\n\nj = dz = *J ; z is the scalar field. dj = 0 = *[div(J)] again.\n\nThe Lagrangian for a particle is the mass; L = - m.\nThe Lagrangian for continua is the mass density ;\n\nL = - n = - sqrt(dz|dz) = - sqrt(j|j)\n\nThen the eqns of motion follow as in my posts.\n\nIf this is the way to do EM, then this is the way to do continuous\nmatter. Its clear to me that this is how to do both theories.\nDoing anything else leads to a\nmess and to errors. For example I do waves in perfect fluids, MHD, and\nplasma in my paper, including the energy and momentum of the waves,\nand the eqns. obeyed by the waves as they propagate, as well as the\nusual dispersion and polarization relations. Everything falls out\neasily if one start from this, the correct framework.\n\nAs I said, Soper published some of this in his book, but no one seems\nto have noticed.\n\nVan Jacques\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>What I am saying in my post is this. A physical theory is based on
Hamilton's principle of least action and canonical field theory. The
problem is

1) what is the field?
2) What is the Lagragian?

Then one just cranks the machinery of classical field theory to get
the Euler Lagrange eqn. and/or the energy momentum tensor (the EMT) T.

First consider EM because people are (strangely) more familiar with EM
than with the theory for (continuous) matter. The basic homogeneous
eqns. for the EM field 2-form F are dF = 0, which give F = dA,
where A is the vector field (actually the 1-form) for the field theory
of EM. We say that F is closed and therefore exact.
There is no guessing or confusion about it, we know how to do this.
We also know that the Lagrangian is

L = - (F|F)/2 + (K|A) ;

Hamilton's principle and canonical field theory then tell us everything
about the EM field; the inhomogeneous eqn. d*F = K
= electric current density, (* = Hodge operator) and the EMT

T = gL - dA \x DL/D(dA)[/itex] (see any text, \x = direct product,

D = gradient operator or covariant derivative,
d = exterior derivative.

This should all be familiar to anyone who has done fields in Landau
and Lifshitz or Jackson.

Now the point of the post. What about continua. What is the
homogenous eqn. which gives the field for continua?
It is the eqn. of continuity. Let j = *J = n*u, where
J = \nu = matter current density, n = (rest) mass density,
and u = 4-velocity. Then write the eqn. of continuity div(J) =
as dj = . The exterior derivative of the 3-form j = 0,
so that j is closed and therefore exact. Because of the isotropy
of 3D space, we have j is the exterior product of the exterior
derivative of 3 scalar fields z^1, z^2, z^3.

j = dz^1 /\ dz^2 /\ dz^3 = *J

so that dj == *[div(J)] and the eqn. of continuity or conservation
of mass is automatically satified.

This becomes even more clear for motion in 1 spatial dimension,
or 2D spacetime. Then both J and j = *J are 1-forms, and

j = dz = *J ; z is the scalar field. dj == *[div(J)] again.

The Lagrangian for a particle is the mass; L = - m.
The Lagrangian for continua is the mass density ;

[itex]L = - n = - \sqrt(dz|dz) = - \sqrt(j|j)

Then the eqns of motion follow as in my posts.

If this is the way to do EM, then this is the way to do continuous
matter. Its clear to me that this is how to do both theories.
Doing anything else leads to a
mess and to errors. For example I do waves in perfect fluids, MHD, and
plasma in my paper, including the energy and momentum of the waves,
and the eqns. obeyed by the waves as they propagate, as well as the
usual dispersion and polarization relations. Everything falls out
easily if one start from this, the correct framework.

As I said, Soper published some of this in his book, but no one seems
to have noticed.

Van Jacques

Van Jacques

Oct25-04, 08:09 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\nArnold Neumaier wrote:\n> Van Jacques wrote:\n>\n> > But what about continuous matter--fluids, plasma, MHD, etc.?\n> > Has this appeared in all the texts while I haven\'t been looking?\n> > If so, can someone give me a ref.?\n> >\n> > Is it thought that matter in space, which makes up our universe,\nthe\n> > galaxies, etc. is not important enough to talk about?\n>\n> Try\n>\n> P. J. Morrison,\n> Hamiltonian description of the ideal fluid,\n> Rev. Mod. Phys., 70, 467--521 (1998).\n\nI would very much like to see this or any other paper on the subject,\nbut only have access through the internet.\nMy searches have turned up nothing on this.\nIs the above paper available on the internet.\n\nI also got a lot of stuff from Marsden\'s site at CalTech, where I was a\ngrad student.\n\nAs far as Goldstein, he fails because the treatment is\nnon-relativistic, and field-theory\nand mechanics are intrinsically relativistic, as they play out in\nspacetime, and\none can\'t just say "we will consider speeds much less than light". More\nserious\nproblems develop, such as the EMT is not symmetric when one tries to do\nit\nnon-relativistically. A relativistic treatment is actually simpler and\nmore consistent,\nesp. if one wants to include the EM field, as in MHD and plasmas, one\ncan\'t have\nnon-rel. matter with an EM field, which is inherently relativistic.\n\nAlso, Goldstien does not introduce the correct fields for continua\n(Soper, Classical\nField Theory is the only place I have found it done correctly--an\nexcellent, but\nlargely ignored book.)\n\nGoldstien calls the positions of the particle x the labels (which I\ncall the fields z^i above)\nand his his field is \\eta = position = my x (most people call the\nposition x).\n\nSo he interchanges the fields and the position. But the main thing is\nthat he does not\nuse the fact that the eqn. of continuity d*J = 0 ==> the 3-form *J is\nclosed and exact.\n\nIts useful to do this in 2D, which makes thing obvious. If one carries\nthru the 2D theory\nof flow starting from d*J = 0 ==> the 1-form *J is closed and exact ==>\n\n*J = dz ; or J = nu = *dz ; where z is a scalar field ,\n\none will see how using the Lagrangian L = - n = sqrt(dz|dz) and the def\nof the\ncanoical EMT gives T = nuu, so div(T) = 0 ==> n du(u) = n a = 0,\n\nwhere a = du(u) = acceleration ; and (u|div(T)) = div(nu) just gives\nback the eq. of continuity.\n\nI have posted on this a few times. I will give the threads in the\nfollowing post.\n\nOf course this can be done in 3D, for perfect fluids, MHD, plasma, and\nmore complex\ncontinua.\n\nVan\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>Arnold Neumaier wrote:
> Van Jacques wrote:
>
> > But what about continuous matter--fluids, plasma, MHD, etc.?
> > Has this appeared in all the texts while I haven't been looking?
> > If so, can someone give me a ref.?
> >
> > Is it thought that matter in space, which makes up our universe,
the
> > galaxies, etc. is not important enough to talk about?
>
> Try
>
> P. J. Morrison,
> Hamiltonian description of the ideal fluid,
> Rev. Mod. Phys., 70, 467--521 (1998).

I would very much like to see this or any other paper on the subject,
but only have access through the internet.
My searches have turned up nothing on this.
Is the above paper available on the internet.

I also got a lot of stuff from Marsden's site at CalTech, where I was a
grad student.

As far as Goldstein, he fails because the treatment is
non-relativistic, and field-theory
and mechanics are intrinsically relativistic, as they play out in
spacetime, and
one can't just say "we will consider speeds much less than light". More
serious
problems develop, such as the EMT is not symmetric when one tries to do
it
non-relativistically. A relativistic treatment is actually simpler and
more consistent,
esp. if one wants to include the EM field, as in MHD and plasmas, one
can't have
non-rel. matter with an EM field, which is inherently relativistic.

Also, Goldstien does not introduce the correct fields for continua
(Soper, Classical
Field Theory is the only place I have found it done correctly--an
excellent, but
largely ignored book.)

Goldstien calls the positions of the particle x the labels (which I
call the fields z^i above)
and his his field is \eta = position = my x (most people call the
position x).

So he interchanges the fields and the position. But the main thing is
that he does not
use the fact that the eqn. of continuity d*J = ==> the 3-form *J is
closed and exact.

Its useful to do this in 2D, which makes thing obvious. If one carries
thru the 2D theory
of flow starting from d*J = ==> the 1-form *J is closed and exact ==>

*J = dz ;[/itex] or J = \nu = *dz ; where z is a scalar field ,

one will see how using the Lagrangian L = - n = \sqrt(dz|dz) and the def
of the
canoical EMT gives T = nuu, so div(T) = ==> n du(u) [itex]= n a = 0,

where a = du(u) = acceleration ; and (u|div(T)) = div(\nu) just gives
back the eq. of continuity.

I have posted on this a few times. I will give the threads in the
following post.

Of course this can be done in 3D, for perfect fluids, MHD, plasma, and
more complex
continua.

Van

Arnold Neumaier

Oct25-04, 10:42 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>\n\nVan Jacques wrote:\n> Arnold Neumaier wrote:\n>\n>>P. J. Morrison,\n>>Hamiltonian description of the ideal fluid,\n>>Rev. Mod. Phys., 70, 467--521 (1998).\n>\n> I would very much like to see this or any other paper on the subject,\n> but only have access through the internet.\n> My searches have turned up nothing on this.\n> Is the above paper available on the internet.\n\nOnly if you have a subscription to Rev. Mod. Phys.\nYou cannot expect to get everything free...\n\nBut there are some related papers:\npeaches.ph.utexas.edu/ifs/ifsreports/825_padhye.pdf\npeaches.ph.utexas.edu/ifs/ifsreports/956_Morrison.pdf\npeaches.ph.utexas.edu/ifs/ ifsreports/Ham_des_shear_flow.pdf\n\n\nArnold Neumaier\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>Van Jacques wrote:
> Arnold Neumaier wrote:
>
>>P. J. Morrison,
>>Hamiltonian description of the ideal fluid,
>>Rev. Mod. Phys., 70, 467--521 (1998).
>
> I would very much like to see this or any other paper on the subject,
> but only have access through the internet.
> My searches have turned up nothing on this.
> Is the above paper available on the internet.

Only if you have a subscription to Rev. Mod. Phys.
You cannot expect to get everything free...

But there are some related papers:
peaches.ph.utexas.edu/ifs/ifsreports/825_padhye.pdf
peaches.ph.utexas.edu/ifs/ifsreports/956_Morrison.pdf
peaches.ph.utexas.edu/ifs/ ifsreports/Ham_des_shear_flow.pdf

Arnold Neumaier

Van Jacques

Oct27-04, 10:56 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> On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:\n\n> > I recall Goldstein\'s "Classical Mechanics" does Lagrangian and\n> > Hamiltonian mechanics for particles, then tries but fails to do\n> > continua in the last chapter--a rather pathetic finish to an\n> > otherwise great text. One can now\n> > include continua and use differential geometry.\n\n> I\'m not sure how Goldstein "fails to do continua" in the last\n> chapter.\n> I don\'t believe the treatment is erroneous, but it is brief\n> compared to the amount of space devoted to other topics.\n\nGoldstein fails in two respects:\nSince one wants to be able to introduce the EM field interacting\nwith (continuous) matter in MHD and plasmas, one must treat everything\nrelativistically (and have Lorentz invariance) from the start.\nNon-relativistic treatments create many problems,\nmaking the introduction of the EM field almost impossible,\nsince the EM field is intrinsically relativistic. I wonder what\nphysicists were thinking when trying to create Newtonian theories\nof MHD and plasmas. You can\'t mix one group of eqns. which are\nLorentz invariant with another group that is Galilean invariant\nwithout creating a mess and introducing serious errors at the\nstart.\n> I\'ve found\n> that the use of differential geometry in physics is dictated more\n> by the culture of a given subfield instead of whatever is\n> fashionable in modern mathematics. Continuum mechanics can be\n> done quite well with the more orthodox use of vector analysis. In\n> fact, this is what is done in most texts on fluid dynamics and\n> elasticity theory (for example the classic texts by Landau and\n> Lifshitz).\n\n\nI thought that the tools of differential geometry and forms\nwere well established by now. They included vector analysis,\nbut they clarify everything. Esp. Stokes thm. and multiple integrals,\nwhich are especially important in dealing with continua.\n\nI gave the example of the EM field, which is a mess without\nthe use of forms. Writing Maxwell\'s eqns. in 4D as\ndF = 0, F = dA, and d*F = *J, and p dimensional integrals over\np-forms should be standard. I hope things like curl(A) and div(B)\nhave been abandoned in favor of dA and *d*B. We can always\nproject onto a 3D subspace of spacetime if necessary.\n\n> > I have posted this before to a deafening silence.\n>\n> Unfortunately, sometimes questions like "What do you think?"\n> are answered implicitly in this way.\n\n\nYes. Or else the posts aren\'t read.\n\n> > The mechanics of continua is not a sexy topic, but it is basic,\n> > like EM and particle motion.\n>\n> I would agree. However, it seems that over the past few decades\n> continuum mechanics has left the regular curriculum of\n> undergraduate\n> physics. This subject seems to have moved over to mechanical\n> engineering. And because people tend to use what they know,\n> continuum\n> mechanics seems to have lost influence over the thoughts of those\n> now engaged in research.\n\n\nWhere ever it is, the basic physics--the fields and the field\ntheory, are still up to physicist to formulate. I think this\nis how things got swept away just when the basic theory was\nreaching solid ground as a field theory, to sit beside EM and\nparticle motion as part of the basic understanding of how the\nworld works. There is also the rich field of waves and the\ninteraction of continua with an EM field left to be put on\na solid foundation. Unfortunately, everyone is so busy with\nparticle theory and black holes (which are important and interesting)\nthat the less sexy, but in a way more important area of the\ntheory of continuous matter, waves, and the interaction with\nthe EM field, has been abandoned to applied physics, as you say.\n\nThe theory has not been put on solid ground as EM has, even\nthough it is analogous to EM.\n\n> > When I learned EM as a classical field theory from L&L , Jackson,\n> > and Misner,Thorne and Wheeler (MTW), I was impressed, as\n> > I think everyone is.\n> >\n> > dF = 0 ==> F = dA, L = - (F|F)/2 + (J|A) ==>\n> >\n> > d*F = 0 and T = gL - dA \\x dL/D(dA) = canonical EMT for the EM\nfield.\n> >\n> > But what about continuous matter--fluids, plasma, MHD, etc.?\n> > Has this appeared in all the texts while I haven\'t been looking?\n> > If so, can someone give me a ref.?\n> > I have been out of touch for some years, have things changed?\n> > Isn\'t field theory--the fields like A or z^i and the Lagrangians of\nany\n> > importance any more? I thought they were basic. This is the best,\n> > and really the only way for advanced work on motion, waves, and\n> > any further study of fluids, plasma, and MHD--all continua.\n> > I know everyone is busy with their own work, but I would be\ngrateful\n> > for any response to the following.\n>\n> I know little about these topics, but from what I\'ve seen the\nresearch\n> on these topics is still active but segregated into its own\ncommunity.\n> That would explain why one does not hear about it very often.\n>\n> > Does everyone now know how to do continua as a field theory?\n> >\n> > d*J = 0 ==> *J = n*u = dz^1 /\\ dz^2 /\\ dz^3 ;\n> >\n> > so the eqn. of continuity implies the existence of the 3 scalar\nfields\n> > z^i for continua?\n> >\n> > Then classical field theory gives the EMT and eqns. of motion:\n> >\n> > T = gL - dz^i dL/dz^i ; where for cold matter L = - n =\nsqrt(*J|*J).\n> >\n> > (The above is valid in GR, but its best to do it for SR first).\n> > Does anyone have any comments on this. The same can be done\n> > for MHD, plasma, or any matter that obeys the conservation of mass\n> > d*J = 0.\n> >\n> > Is it too basic too talk about? Is this well known now. It wasn\'t\n> > known at all when I learned physics, and I have seen no changes\nsince\n> > then, except advances in particle theory.\n>\n> Although I still find your equations a little hard to decipher,\n\nWhat is hard to decipher? d = exterior derivative,\n/\\ = exterior product. I use \\x to denote the direct product in\nthe EMT, but if you are familiar with classical field theory\nyou should know how T is calculated from the Lagrangian.\nI will leave out \\x as I guess it doesn\'t help.\nI assume that one has been thru the field theory of the EM\nfield. If not, this post won\'t make sense, but all physicists\nshould be familiar with that.\n\n>I think all of this is well known by now, even if it is usually\n>expressed in different language.\n\nIs it? The only place I have seen it expressed in any language is in\nSoper\'s "Classical Field Theory", and he left a lot to be done.\n\n> Even if there has been some advance in this field,\n> it would hardly have made it to the mainstream presses. After all,\n> underlying this theory are just basic Newton\'s laws of motion.\n> If you are interested in how fluid dynamics is done in a relativistic\n\n> setting, I think astrophysics is the right place to look for the\n> literature. Otherwise, most fluid dynamics work that I\'ve seen is\nquite\n> non-relativistic.\n\n\nYes, unfortunately. The same is true for MHD and plasmas, which\nmakes them nonsense. Also, all these need to be done as classical\nfield theories, where it is obvious what the fields are, and\nno one can say "why did you use these as the fields", any more\nthan they could say "why is A the field for the field theory of\nEM". d*J = 0 <==> *J = 3 scalar fields as above, and\n\n> > I would appreciate some comment from anyone.\n>\n> I\'ve learned that formulating a theory is rarely enough to attract\n> attention. The theory must be applied to solve some problems. I don\'t\n\n> know if there would be an audience for your paper. But if you wish to\n\n> talk about it, you could post here some worked out problems using\nthis\n> theory. I\'m sure it would be enlightening and a nice exercise in\n> differential geometry and continuum mechanics for anyone who follows.\n\n\n\nThe point I am making is that the theory is much more powerful\nin the field theoretic form. This is how things should be done\nin modern physics--it is how classical EM is done.\nThe knowledge that A is the vector field for the field theory\nof EM, and that F = dA, and L = - (F|F)/2 is important and\na powerful tool for problems in EM, as well as for integrating\nit with the rest of physics and for making progress.\n\nThe same is true for continua. Everyone should know that the\nfield theory of continua has 3 scalar fields, the Lagrangian\ncoordinates, which come from the eqn. of continuity d*J = 0,\nand the Lagrangian L = - r + L_em,\n\nwhere r = total energy density, and L_em = Lagrangian for\nany EM fields.\n\nI carry all this thru and apply it to wave propagation in\nmy paper.\nVan\n> Igor\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 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:

> > I recall Goldstein's "Classical Mechanics" does Lagrangian and
> > Hamiltonian mechanics for particles, then tries but fails to do
> > continua in the last chapter--a rather pathetic finish to an
> > otherwise great text. One can now
> > include continua and use differential geometry.

> I'm not sure how Goldstein "fails to do continua" in the last
> chapter.
> I don't believe the treatment is erroneous, but it is brief
> compared to the amount of space devoted to other topics.

Goldstein fails in two respects:
Since one wants to be able to introduce the EM field interacting
with (continuous) matter in MHD and plasmas, one must treat everything
relativistically (and have Lorentz invariance) from the start.
Non-relativistic treatments create many problems,
making the introduction of the EM field almost impossible,
since the EM field is intrinsically relativistic. I wonder what
physicists were thinking when trying to create Newtonian theories
of MHD and plasmas. You can't mix one group of eqns. which are
Lorentz invariant with another group that is Galilean invariant
without creating a mess and introducing serious errors at the
start.
> I've found
> that the use of differential geometry in physics is dictated more
> by the culture of a given subfield instead of whatever is
> fashionable in modern mathematics. Continuum mechanics can be
> done quite well with the more orthodox use of vector analysis. In
> fact, this is what is done in most texts on fluid dynamics and
> elasticity theory (for example the classic texts by Landau and
> Lifshitz).

I thought that the tools of differential geometry and forms
were well established by now. They included vector analysis,
but they clarify everything. Esp. Stokes thm. and multiple integrals,
which are especially important in dealing with continua.

I gave the example of the EM field, which is a mess without
the use of forms. Writing Maxwell's eqns. in 4D as
dF = 0, F = dA, and d*F = *J, and p dimensional integrals over
p-forms should be standard. I hope things like curl(A) and div(B)
have been abandoned in favor of dA and *d*B. We can always
project onto a 3D subspace of spacetime if necessary.

> > I have posted this before to a deafening silence.
>
> Unfortunately, sometimes questions like "What do you think?"
> are answered implicitly in this way.

Yes. Or else the posts aren't read.

> > The mechanics of continua is not a sexy topic, but it is basic,
> > like EM and particle motion.
>
> I would agree. However, it seems that over the past few decades
> continuum mechanics has left the regular curriculum of
> undergraduate
> physics. This subject seems to have moved over to mechanical
> engineering. And because people tend to use what they know,
> continuum
> mechanics seems to have lost influence over the thoughts of those
> now engaged in research.

Where ever it is, the basic physics--the fields and the field
theory, are still up to physicist to formulate. I think this
is how things got swept away just when the basic theory was
reaching solid ground as a field theory, to sit beside EM and
particle motion as part of the basic understanding of how the
world works. There is also the rich field of waves and the
interaction of continua with an EM field left to be put on
a solid foundation. Unfortunately, everyone is so busy with
particle theory and black holes (which are important and interesting)
that the less sexy, but in a way more important area of the
theory of continuous matter, waves, and the interaction with
the EM field, has been abandoned to applied physics, as you say.

The theory has not been put on solid ground as EM has, even
though it is analogous to EM.

> > When I learned EM as a classical field theory from L&L , Jackson,
> > and Misner,Thorne and Wheeler (MTW), I was impressed, as
> > I think everyone is.
> >
> > dF = ==> F = dA, L = - (F|F)/2 + (J|A) ==>
> >
> > d*F = and T = gL - dA \x dL/D(dA) = canonical EMT for the EM
field.
> >
> > But what about continuous matter--fluids, plasma, MHD, etc.?
> > Has this appeared in all the texts while I haven't been looking?
> > If so, can someone give me a ref.?
> > I have been out of touch for some years, have things changed?
> > Isn't field theory--the fields like A or z^i and the Lagrangians of
any
> > importance any more? I thought they were basic. This is the best,
> > and really the only way for advanced work on motion, waves, and
> > any further study of fluids, plasma, and MHD--all continua.
> > I know everyone is busy with their own work, but I would be
grateful
> > for any response to the following.
>
> I know little about these topics, but from what I've seen the
research
> on these topics is still active but segregated into its own
community.
> That would explain why one does not hear about it very often.
>
> > Does everyone now know how to do continua as a field theory?
> >
> > d*J = ==> *J = n*u = dz^1 /\ dz^2 /\ dz^3 ;
> >
> > so the eqn. of continuity implies the existence of the 3 scalar
fields
> > z^i for continua?
> >
> > Then classical field theory gives the EMT and eqns. of motion:
> >
> > T = gL - dz^i dL/dz^i ; where for cold matter L = - n =\sqrt(*J|*J).
> >
> > (The above is valid in GR, but its best to do it for SR first).
> > Does anyone have any comments on this. The same can be done
> > for MHD, plasma, or any matter that obeys the conservation of mass
> > d*J = .
> >
> > Is it too basic too talk about? Is this well known now. It wasn't
> > known at all when I learned physics, and I have seen no changes
since
> > then, except advances in particle theory.
>
> Although I still find your equations a little hard to decipher,

What is hard to decipher? d = exterior derivative,
/\ = exterior product. I use \x to denote the direct product in
the EMT, but if you are familiar with classical field theory
you should know how T is calculated from the Lagrangian.
I will leave out \x as I guess it doesn't help.
I assume that one has been thru the field theory of the EM
field. If not, this post won't make sense, but all physicists
should be familiar with that.

>I think all of this is well known by now, even if it is usually
>expressed in different language.

Is it? The only place I have seen it expressed in any language is in
Soper's "Classical Field Theory", and he left a lot to be done.

> Even if there has been some advance in this field,
> it would hardly have made it to the mainstream presses. After all,
> underlying this theory are just basic Newton's laws of motion.
> If you are interested in how fluid dynamics is done in a relativistic

> setting, I think astrophysics is the right place to look for the
> literature. Otherwise, most fluid dynamics work that I've seen is
quite
> non-relativistic.

Yes, unfortunately. The same is true for MHD and plasmas, which
makes them nonsense. Also, all these need to be done as classical
field theories, where it is obvious what the fields are, and
no one can say "why did you use these as the fields", any more
than they could say "why is A the field for the field theory of
EM". d*J = <==> *J = 3 scalar fields as above, and

> > I would appreciate some comment from anyone.
>
> I've learned that formulating a theory is rarely enough to attract
> attention. The theory must be applied to solve some problems. I don't

> know if there would be an audience for your paper. But if you wish to

> talk about it, you could post here some worked out problems using
this
> theory. I'm sure it would be enlightening and a nice exercise in
> differential geometry and continuum mechanics for anyone who follows.

The point I am making is that the theory is much more powerful
in the field theoretic form. This is how things should be done
in modern physics--it is how classical EM is done.
The knowledge that A is the vector field for the field theory
of EM, and that F = dA, and L = - (F|F)/2 is important and
a powerful tool for problems in EM, as well as for integrating
it with the rest of physics and for making progress.

The same is true for continua. Everyone should know that the
field theory of continua has 3 scalar fields, the Lagrangian
coordinates, which come from the eqn. of continuity d*J = 0,
and the Lagrangian L = - r + L_{em},

where r = total energy density, and L_{em} = Lagrangian for
any EM fields.

I carry all this thru and apply it to wave propagation in
my paper.
Van
> Igor

Van Jacques

Oct27-04, 10:56 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\nArnold Neumaier wrote:\n> Van Jacques wrote:\n> > Arnold Neumaier wrote:\n> >\n> >>P. J. Morrison,\n> >>Hamiltonian description of the ideal fluid,\n> >>Rev. Mod. Phys., 70, 467--521 (1998).\n> >\n> > I would very much like to see this or any other paper on the\nsubject,\n> > but only have access through the internet.\n> > My searches have turned up nothing on this.\n> > Is the above paper available on the internet.\n>\n> Only if you have a subscription to Rev. Mod. Phys.\n> You cannot expect to get everything free...\n>\n> But there are some related papers:\n> peaches.ph.utexas.edu/ifs/ifsreports/825_padhye.pdf\n> peaches.ph.utexas.edu/ifs/ifsreports/956_Morrison.pdf\n> peaches.ph.utexas.edu/ifs/ ifsreports/Ham_des_shear_flow.pdf\n========\nThanks for the URLs. I got them and looked at them, and\nplan to email the authors about their work.\n\nThe secret to the whole problem, to seeing how things work,\nis to do the problem relativistically. Any attempt at a\nnon-rel. soln. will lead only to confusion and a mess.\nIts just like EM theory, with dF = 0, F = dA.\n\nIt took a while to do it using div(E) = 0 and\ncurl(B) + dE/dt = 0 (d/dt = partial deriv), and\n\nE = - grad(A) + d(\\phi)/dt ; B = curl(A).\n\nClearly the 1st way clearer and more powerful.\nAt least both are relativistic. The use of 4D notation\nrather than 3 + 1 makes things a lot clearer, however.\n\nThe same is true for continua, and MHD and plasma, although\nthey have the added error of usually being done\nnon-relativistically.\n\nOne _must_ use 4D spacetime, and do things relativistically,\nor you won\'t see the essential relationships which are\nthere to be seen. You must write the eq. of continuity first as\n\ndiv(J) = div(nu) = 0 ; where div is the 4D operator,\n\nJ = nu is the 4-vector matter current,\n\nJ^0 = ng, J^i = ngv^i ; n = proper density (Lorentz inv.)\n\ng = 1/sqrt(1 - v^2) ; v^i = dx^i/dt = 3-velocity.\n\nThen one must write div = *d*, so that\n\ndiv(J) = 0 <==> *d*J = 0 <==> dj = 0 ; where j = *J is\n\nthe dual to the matter current J. This is the only\nway to see what is going on. Older approaches lead to\nconfusion and inability to see what is happening.\n\nFor example, how much matter is in a 3D region R?\n\nInt_R(*J) = N = number of particles in R = rest mass in R.\n\nThe volume element 3-form is part of the current, as it should\nbe. R could be any 3D subspace of 4D, e.g., the flux of matter\nthru a 2-sphere per unit time. This works esp. in EM for F,\nif you have ever done EM problems with F, the EM field 2-form.\n\nThen the Lagrangian is not non-relativistic\n\nL = \\rho v^2/2 - \\phi(x,t) (as in the refs. above--and god\n\nknows what the fields used are), but\n\nL = - n = - sqrt(*J|*J) ; where *J = dz^1 /\\ dz^2 /\\ dz^3.\n\nThis is absolutely fundamental to understanding any continua.\nIf you don\'t know what the fields are, you are in trouble.\n\nAnd classical field theory falls out naturally, canonically\n(i.e. "by the law"). We have the EMT\n\nT = gL - dz^i dL/D(dz^i) ; as one always has in canonical field theory.\n(g = diag(-1,1,1,1) = metric of flat spacetime).\n\ndiv(T) = 0 give the eqns. of motion and conservation of energy.\n\n(If the expression for T is confusing, it is hard with ASCII,\nthough I could write it out in components using TeX notation;\n\nT^i_j = L g^i_j - z^k_j dL/dz^k_i ; where z^k_j = dz^k/dx^j\n\nk = 1,2,3; (i,j) = 0,1,2,3\n\nI don\'t want to be critical of other\'s work, but\nI believe there is room for\nimprovement using the ideas I present here.\n\nI will email them, but experience has taught me that by\nno longer working at a university I have lost my voice.\nI believe these ideas are important.\n\nVan\n>\n>\n> Arnold Neumaier\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>Arnold Neumaier wrote:
> Van Jacques wrote:
> > Arnold Neumaier wrote:
> >
> >>P. J. Morrison,
> >>Hamiltonian description of the ideal fluid,
> >>Rev. Mod. Phys., 70, 467--521 (1998).
> >
> > I would very much like to see this or any other paper on the
subject,
> > but only have access through the internet.
> > My searches have turned up nothing on this.
> > Is the above paper available on the internet.
>
> Only if you have a subscription to Rev. Mod. Phys.
> You cannot expect to get everything free...
>
> But there are some related papers:
> peaches.ph.utexas.edu/ifs/ifsreports/825_padhye.pdf
> peaches.ph.utexas.edu/ifs/ifsreports/956_Morrison.pdf
> peaches.ph.utexas.edu/ifs/ ifsreports/Ham_des_shear_flow.pdf
========
Thanks for the URLs. I got them and looked at them, and
plan to email the authors about their work.

The secret to the whole problem, to seeing how things work,
is to do the problem relativistically. Any attempt at a
non-rel. soln. will lead only to confusion and a mess.
Its just like EM theory, with dF = 0, F = dA.

It took a while to do it using div(E) = and
curl(B) + dE/dt =(d/dt = partial deriv), and

E = - grad(A) + d(\phi)/dt ; B =[/itex] curl(A).

Clearly the 1st way clearer and more powerful.
At least both are relativistic. The use of 4D notation
rather than 3 + 1 makes things a lot clearer, however.

The same is true for continua, and MHD and plasma, although
they have the added error of usually being done
non-relativistically.

One _must_ use 4D spacetime, and do things relativistically,
or you won't see the essential relationships which are
there to be seen. You must write the eq. of continuity first as

div(J) = div(\nu) = ; where div is the 4D operator,

J = \nu is the 4-vector matter current,

J^0 = ng, J^i = ngv^i ; n = proper density (Lorentz inv.)

g = 1/\sqrt(1 - v^2) ; v^i = dx^i/dt = 3-velocity.

Then one must write div = *d*, so that

div(J) = <==> *d*J = <==> dj = ; where j = *J is

the dual to the matter current J. This is the only
way to see what is going on. Older approaches lead to
confusion and inability to see what is happening.

For example, how much matter is in a 3D region R?

\Int_R(*J) = N = number of particles in R = rest mass in R.

The volume element 3-form is part of the current, as it should
be. R could be any 3D subspace of 4D, e.g., the flux of matter
thru a 2-sphere per unit time. This works esp. in EM for F,
if you have ever done EM problems with F, the EM field 2-form.

Then the Lagrangian is not non-relativistic

L = \rho v^2/2 - \phi(x,t) (as in the refs. above--and god

knows what the fields used are), but

L = - n = - \sqrt(*J|*J) ; where *J = dz^1 /\ dz^2 /\ dz^3.

This is absolutely fundamental to understanding any continua.
If you don't know what the fields are, you are in trouble.

And classical field theory falls out naturally, canonically
(i.e. "by the law"). We have the EMT

T = gL - dz^i dL/D(dz^i) ; as one always has in canonical field theory.
(g = diag(-1,1,1,1) = metric of flat spacetime).

div(T) = give the eqns. of motion and conservation of energy.

(If the expression for T is confusing, it is hard with ASCII,
though I could write it out in components using TeX notation;

T^{i_j} = L g^{i_j} - z^{k_j} dL/dz^k_i ; where [itex]z^{k_j} = dz^k/dx^jk = 1,2,3; (i,j) = 0,1,2,3

I don't want to be critical of other's work, but
I believe there is room for
improvement using the ideas I present here.

I will email them, but experience has taught me that by
no longer working at a university I have lost my voice.
I believe these ideas are important.

Van
>
>
> Arnold Neumaier

Igor Khavkine

Oct27-04, 11:00 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\'ve mentioned in a previous post that I have trouble interpreting your\nnotation and conclusions. Let me try to interpret what I understand and\nperhaps you can fill in some blanks.\n\nOn Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:\n\n> Now the point of the post. What about continua. What is the homogenous\n> eqn. which gives the field for continua? It is the eqn. of continuity. Let\n> j = *J = n*u, where J = nu = matter current density, n = (rest) mass\n> density, and u = 4-velocity.\n\nThe first point is that I do not think it is a trivial fact that\ncurrent density is represented by a 3-form in 4D spacetime. Second, I am\nconfused by your use of the Hodge star. You seem to be claiming that J\nis a 1-form which is proportional to u, where u is a 4-velocity vector.\nI guess the identification can be made with the metric, but this step\ncannot be omitted.\n\nFor my own benefit and that of anyone interested, let me sketch that a\ncurrent density in an (k+1)-dimensional spacetime must indeed be an\nk-form. Start off in familiar territory and assume that space and time\nare split. Let n\' be the particle density (n\' integrated over a k-volume\nof space gives a number, must be a k-form on space). Let j\' be the particle\ncurrent density. Given a (k-1)-dimensional hypersurface, the integral of\nj\' over it also gives a number, the flux through the hypersurface per\nunit time. Hence j\' must be a (k-1)-form on space. The continuity\nequation takes the form (where V is a k-volume of space, and @V its\nboundary)\n\n@/@t int_V n = int_(@V) j\n\nor equilvalently in integral form\n\nint_(V at t2) n\' - int_(V at t1) n\' = int_(t1 to t2) int_(@V) j\' dt\n= int_(@Vx[t1,t2]) j\' /\\ dt .\n\nNow we have two k-forms on spacetime, n\' and j\' /\\ dt, let me denote\ntheir sum by j = n\' + j\' /\\ dt. Looking at the above equation carefully,\nwe note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the\nboundary of the spacetime volume Vx[t1,t2]. Now the continuity equation\ncan be rewritten as\n\nint_@(Vx[t1,t2]) j = 0 which implies int_(Vx[t1,t2]) dj = 0.\n\nAnd since I can build any spacetime volume out of little pieces that\nlook like Vx[t1,t2], the differential form of the continuity equation\nis just dj = 0.\n\nThe above discussion justifies two things. First that the current\ndensity is given by a k-form j in a (k+1)-dimensional space time.\nIntegrated over a space-like k-volume, j gives the number of particles\ncontained in it. While integrated over a timelike k-hypersurface, j\ngives the total number of particles that have passed through it. Second\nit justifies the simple dj = 0 form of the continuity equation in\nspace-time. I might have messed up with the sign of the second term in\nj, perhaps it is dt /\\ j, but I think the general idea is fine.\n\nNow, about the relation between the velocity field u and the current\ndensity j. They can\'t be simply related by a Hodge transformation, since\none is a form and the other is a vector field. And how does the\napplication of the Hodge star follow from first principles anyway?\n\n> Then write the eqn. of continuity div(J) = 0\n> as dj = 0. The exterior derivative of the 3-form j = 0, so that j is\n> closed and therefore exact. Because of the isotropy of 3D space, we have j\n> is the exterior product of the exterior derivative of 3 scalar fields z^1,\n> z^2, z^3.\n>\n> j = dz^1 /\\ dz^2 /\\ dz^3 = *J\n\nI don\'t understand your last comment. How does "isotropy of 3D space"\nlead to this particular form of j? By analogy with E&M, and by Poicare\'s\nlemma, since dj = 0, then j = dL for some (k-1)-form A on spacetime.\nWhat would this (k-1)-form dL represent?\n\n> If this is the way to do EM, then this is the way to do continuous matter.\n> Its clear to me that this is how to do both theories. Doing anything else\n> leads to a\n> mess and to errors. For example I do waves in perfect fluids, MHD, and\n> plasma in my paper, including the energy and momentum of the waves, and\n> the eqns. obeyed by the waves as they propagate, as well as the usual\n> dispersion and polarization relations. Everything falls out easily if one\n> start from this, the correct framework.\n\nAs far as I know, people have been doing hydrodynamics before\ndifferential geometry in its current for was formulated. So notation and\nformalism are definitely not an obstacle for theoretical physics. It may\nbe true that using the notation you suggest it is easier to get all the\nequations of motion and such, but it is certainly not the only way. The\nmost probably reason for this formalism not being used in hydrodynamics\nis that people who are doing the latter are usually not familiar with\nthe latter.\n\n> As I said, Soper published some of this in his book, but no one seems to\n> have noticed.\n\nMost probably so. And they probably never will unless someone points it\nout along with definite advantages to introducing a new formalism in\ntheir work.\n\nIgor\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've mentioned in a previous post that I have trouble interpreting your
notation and conclusions. Let me try to interpret what I understand and
perhaps you can fill in some blanks.

On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:

> Now the point of the post. What about continua. What is the homogenous
> eqn. which gives the field for continua? It is the eqn. of continuity. Let
> j = *J = n*u, where J = \nu = matter current density, n = (rest) mass
> density, and u = 4-velocity.

The first point is that I do not think it is a trivial fact that
current density is represented by a 3-form in 4D spacetime. Second, I am
confused by your use of the Hodge star. You seem to be claiming that J
is a 1-form which is proportional to u, where u is a 4-velocity vector.
I guess the identification can be made with the metric, but this step
cannot be omitted.

For my own benefit and that of anyone interested, let me sketch that a
current density in an (k+1)-dimensional spacetime must indeed be an
k-form. Start off in familiar territory and assume that space and time
are split. Let n' be the particle density (n' integrated over a k-volume
of space gives a number, must be a k-form on space). Let j' be the particle
current density. Given a (k-1)-dimensional hypersurface, the integral of
j' over it also gives a number, the flux through the hypersurface per
unit time. Hence j' must be a (k-1)-form on space. The continuity
equation takes the form (where V is a k-volume of space, and @V its
boundary)

@/@t \int_V n = \int_(@V) j

or equilvalently in integral form

\int_(V at t2) n' - \int_(V at t1) n' = \int_(t1 to t2) \int_(@V) j' dt= \int_(@Vx[t1,t2]) j' /\ dt .

Now we have two k-forms on spacetime, n' and j' /\ dt, let me denote
their sum by j = n' + j' /\ dt. Looking at the above equation carefully,
we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the
boundary of the spacetime volume Vx[t1,t2]. Now the continuity equation
can be rewritten as

\int_@(Vx[t1,t2]) j =[/itex] which implies \int_(Vx[t1,t2]) dj = .

And since I can build any spacetime volume out of little pieces that
look like Vx[t1,t2], the differential form of the continuity equation
is just dj = .

The above discussion justifies two things. First that the current
density is given by a k-form j in a (k+1)-dimensional space time.
Integrated over a space-like k-volume, j gives the number of particles
contained in it. While integrated over a timelike k-hypersurface, j
gives the total number of particles that have passed through it. Second
it justifies the simple dj = form of the continuity equation in
space-time. I might have messed up with the sign of the second term in
j, perhaps it is dt /\ j, but I think the general idea is fine.

Now, about the relation between the velocity field u and the current
density j. They can't be simply related by a Hodge transformation, since
one is a form and the other is a vector field. And how does the
application of the Hodge star follow from first principles anyway?

> Then write the eqn. of continuity div(J) =
> as dj = . The exterior derivative of the 3-form j = 0, so that j is
> closed and therefore exact. Because of the isotropy of 3D space, we have j
> is the exterior product of the exterior derivative of 3 scalar fields z^1,
> z^2, z^3.
>
> [itex]j = dz^1 /\ dz^2 /\ dz^3 = *J

I don't understand your last comment. How does "isotropy of 3D space"
lead to this particular form of j? By analogy with E&M, and by Poicare's
lemma, since dj = 0, then j = dL for some (k-1)-form A on spacetime.
What would this (k-1)-form dL represent?

> If this is the way to do EM, then this is the way to do continuous matter.
> Its clear to me that this is how to do both theories. Doing anything else
> leads to a
> mess and to errors. For example I do waves in perfect fluids, MHD, and
> plasma in my paper, including the energy and momentum of the waves, and
> the eqns. obeyed by the waves as they propagate, as well as the usual
> dispersion and polarization relations. Everything falls out easily if one
> start from this, the correct framework.

As far as I know, people have been doing hydrodynamics before
differential geometry in its current for was formulated. So notation and
formalism are definitely not an obstacle for theoretical physics. It may
be true that using the notation you suggest it is easier to get all the
equations of motion and such, but it is certainly not the only way. The
most probably reason for this formalism not being used in hydrodynamics
is that people who are doing the latter are usually not familiar with
the latter.

> As I said, Soper published some of this in his book, but no one seems to
> have noticed.

Most probably so. And they probably never will unless someone points it
out along with definite advantages to introducing a new formalism in
their work.

Igor

Igor Khavkine

Oct28-04, 01:28 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>\n\nOn Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:\n\n>\n>> On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:\n>\n>> > I recall Goldstein\'s "Classical Mechanics" does Lagrangian and\n>> > Hamiltonian mechanics for particles, then tries but fails to do\n>> > continua in the last chapter--a rather pathetic finish to an otherwise\n>> > great text. One can now\n>> > include continua and use differential geometry.\n>\n>> I\'m not sure how Goldstein "fails to do continua" in the last chapter.\n>> I don\'t believe the treatment is erroneous, but it is brief compared to\n>> the amount of space devoted to other topics.\n>\n> Goldstein fails in two respects:\n> Since one wants to be able to introduce the EM field interacting with\n> (continuous) matter in MHD and plasmas, one must treat everything\n> relativistically (and have Lorentz invariance) from the start.\n> Non-relativistic treatments create many problems, making the introduction\n> of the EM field almost impossible, since the EM field is intrinsically\n> relativistic. I wonder what physicists were thinking when trying to create\n> Newtonian theories of MHD and plasmas. You can\'t mix one group of eqns.\n> which are Lorentz invariant with another group that is Galilean invariant\n> without creating a mess and introducing serious errors at the start.\n\nI don not think that doing magnetohydrodynamics was one of Goldstein\'s\ngoals for writing that chapter. Neither is his goal to do an exposition of\ncontinuum mechanics. As best I can tell, the intent of the last chapter of\nGoldstein\'s book is to introduce the reader to the generalities of the\nformulation of classical field theory, as well as introduce some classical\nfields that will later be used in relativistic quantum mechanics and QFT.\n\nAlso, referring to your other post discussing Goldstein, I can only say\nthat his choice of notation has nothing to do with the validity of what he\nis trying to say. And as far as I know, his treatment contains no serious\nerrors. Also, his use of index and vector notation as opposed to\nthe more modern index-less one is no more a draw back than the use of the\nsame notation for treating classical mechanics of point particles.\n\n>> I\'ve found\n>> that the use of differential geometry in physics is dictated more by the\n>> culture of a given subfield instead of whatever is fashionable in modern\n>> mathematics. Continuum mechanics can be done quite well with the more\n>> orthodox use of vector analysis. In fact, this is what is done in most\n>> texts on fluid dynamics and elasticity theory (for example the classic\n>> texts by Landau and Lifshitz).\n>\n> I thought that the tools of differential geometry and forms were well\n> established by now. They included vector analysis, but they clarify\n> everything. Esp. Stokes thm. and multiple integrals, which are especially\n> important in dealing with continua.\n>\n> I gave the example of the EM field, which is a mess without the use of\n> forms. Writing Maxwell\'s eqns. in 4D as dF = 0, F = dA, and d*F = *J, and\n> p dimensional integrals over p-forms should be standard. I hope things\n> like curl(A) and div(B) have been abandoned in favor of dA and *d*B. We\n> can always project onto a 3D subspace of spacetime if necessary.\n\nI would disagree with your assumption. Basic differential geometry is not\npart of the regular undergraduate curriculum for pretty much any\ndiscipline, except perhaps mathematics. Vector analysis is, but it is hard\nto make the connection between the two without prior exposure.\n\nI also take issue with your claim that since modern diff.geo. notation is\nmore compact and more elegant that it is more powerful. First of all,\n"powerful" is not a well defined term. Instead one can look at its\nutility. But before discussing utility, a specific purpose must be stated.\nIf the purpose is to express all the dynamical equations of your theory as\ncompactly and elegantly as possible? Then yes, I\'d say it\'s useful. If the\npurpose is to perform a calculation in some geometry where space and time\nare naturally split, then I\'d say that regular vector analysis is more\nuseful. If the purpose is to perform some numerical simulation, then\nexplicit coordinate choices must be made and everything must be calculated\nin components. For this case I\'d say that the abstract index-less\nformalism is one of the most useless.\n\nIn conclusion, is modern diff.geo. notation universally useful? No. Is it\nuniversally useless? Also no. Just as many other things, it falls\nsomewhere in between. To advocate the use of one notation or formalism\nover another, the intended purpose must be stated and a strong argument\nfor the advantages for this particular purpose must be made.\n\n>> Although I still find your equations a little hard to decipher,\n>\n> What is hard to decipher? d = exterior derivative, /\\ = exterior product.\n> I use \\x to denote the direct product in the EMT, but if you are familiar\n> with classical field theory you should know how T is calculated from the\n> Lagrangian. I will leave out \\x as I guess it doesn\'t help. I assume that\n> one has been thru the field theory of the EM field. If not, this post\n> won\'t make sense, but all physicists should be familiar with that.\n\nI find your equations hard to decipher because you make many claims\nwithout justification that are not obviously true, although they very well\nmight be. At least they are not obvious to me. I\'ve taken a crack at\nsorting some of your equations out in another branch of this thread, your\nclarifications are welcome.\n\n>>I think all of this is well known by now, even if it is usually expressed\n>>in different language.\n>\n> Is it? The only place I have seen it expressed in any language is in\n> Soper\'s "Classical Field Theory", and he left a lot to be done.\n\nI don not know of where to find a thorough relativistic treatment of\ncontinuum mechanics coupled to E&M. But some of this is worked out in\nChapter IV of Landau\'s Classical Field Theory (vol. 2) and Chapter XV of\nhis Hydrodynamics (vol. 6). Also, take a look at Chapter VIII of\nElectrodynamics of Continuous Media by Landau and Lifshitz (vol. 8). There\nthe equations for non-relativistic MHD are worked out. You have a point\nabout possible pitfalls of using a Galilean theory for matter coupled to a\nLorentzian theory for the E&M field. However, the E&M equations can also\nbe reduced to a set of Galilean equations when the limit c -> oo is taken\ninto account. This amounts to dropping the terms corresponding to the\nelectric displacement current and the Faraday effect.\n\n> The point I am making is that the theory is much more powerful in the\n> field theoretic form. This is how things should be done in modern\n> physics--it is how classical EM is done. The knowledge that A is the\n> vector field for the field theory of EM, and that F = dA, and L = -\n> (F|F)/2 is important and a powerful tool for problems in EM, as well as\n> for integrating it with the rest of physics and for making progress.\n>\n> The same is true for continua. Everyone should know that the field theory\n> of continua has 3 scalar fields, the Lagrangian coordinates, which come\n> from the eqn. of continuity d*J = 0, and the Lagrangian L = - r + L_em,\n>\n> where r = total energy density, and L_em = Lagrangian for any EM fields.\n\nMore powerful? That\'s an ambiguous statement that I discussed above.\nI think it would be a safe guess that anyone working on a particular\ntheory knows what the relevant dynamical variables of that theory are.\nBut the choice of dynamical variables is not unique, you can also do a\nchange of variables. In E&M, all the equations can be expressed either in\nterms of A or F. A is the variable with respect to which the variational\nproblem is constructed, but even then. I could define something like A\' =\n*A, or A\' = A + dV, or A\' = (A|.), or other possibly non-linear field\nredefinitions. Just as with E&M, all hydrodynamical equations can be\nexpressed in terms of the current density j or as you do in terms of the\nLagrangian coordinates, or perhaps Eulerian coordinates or some other\nfield definition. Once again, you must state your goal and argue why your\nchoice of field is best.\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>On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:

>
>> On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:
>
>> > I recall Goldstein's "Classical Mechanics" does Lagrangian and
>> > Hamiltonian mechanics for particles, then tries but fails to do
>> > continua in the last chapter--a rather pathetic finish to an otherwise
>> > great text. One can now
>> > include continua and use differential geometry.
>
>> I'm not sure how Goldstein "fails to do continua" in the last chapter.
>> I don't believe the treatment is erroneous, but it is brief compared to
>> the amount of space devoted to other topics.
>
> Goldstein fails in two respects:
> Since one wants to be able to introduce the EM field interacting with
> (continuous) matter in MHD and plasmas, one must treat everything
> relativistically (and have Lorentz invariance) from the start.
> Non-relativistic treatments create many problems, making the introduction
> of the EM field almost impossible, since the EM field is intrinsically
> relativistic. I wonder what physicists were thinking when trying to create
> Newtonian theories of MHD and plasmas. You can't mix one group of eqns.
> which are Lorentz invariant with another group that is Galilean invariant
> without creating a mess and introducing serious errors at the start.

I don not think that doing magnetohydrodynamics was one of Goldstein's
goals for writing that chapter. Neither is his goal to do an exposition of
continuum mechanics. As best I can tell, the intent of the last chapter of
Goldstein's book is to introduce the reader to the generalities of the
formulation of classical field theory, as well as introduce some classical
fields that will later be used in relativistic quantum mechanics and QFT.

Also, referring to your other post discussing Goldstein, I can only say
that his choice of notation has nothing to do with the validity of what he
is trying to say. And as far as I know, his treatment contains no serious
errors. Also, his use of index and vector notation as opposed to
the more modern index-less one is no more a draw back than the use of the
same notation for treating classical mechanics of point particles.

>> I've found
>> that the use of differential geometry in physics is dictated more by the
>> culture of a given subfield instead of whatever is fashionable in modern
>> mathematics. Continuum mechanics can be done quite well with the more
>> orthodox use of vector analysis. In fact, this is what is done in most
>> texts on fluid dynamics and elasticity theory (for example the classic
>> texts by Landau and Lifshitz).
>
> I thought that the tools of differential geometry and forms were well
> established by now. They included vector analysis, but they clarify
> everything. Esp. Stokes thm. and multiple integrals, which are especially
> important in dealing with continua.
>
> I gave the example of the EM field, which is a mess without the use of
> forms. Writing Maxwell's eqns. in 4D as dF = 0, F = dA, and d*F = *J, and
> p dimensional integrals over p-forms should be standard. I hope things
> like curl(A) and div(B) have been abandoned in favor of dA and *d*B. We
> can always project onto a 3D subspace of spacetime if necessary.

I would disagree with your assumption. Basic differential geometry is not
part of the regular undergraduate curriculum for pretty much any
discipline, except perhaps mathematics. Vector analysis is, but it is hard
to make the connection between the two without prior exposure.

I also take issue with your claim that since modern diff.geo. notation is
more compact and more elegant that it is more powerful. First of all,
"powerful" is not a well defined term. Instead one can look at its
utility. But before discussing utility, a specific purpose must be stated.
If the purpose is to express all the dynamical equations of your theory as
compactly and elegantly as possible? Then yes, I'd say it's useful. If the
purpose is to perform a calculation in some geometry where space and time
are naturally split, then I'd say that regular vector analysis is more
useful. If the purpose is to perform some numerical simulation, then
explicit coordinate choices must be made and everything must be calculated
in components. For this case I'd say that the abstract index-less
formalism is one of the most useless.

In conclusion, is modern diff.geo. notation universally useful? No. Is it
universally useless? Also no. Just as many other things, it falls
somewhere in between. To advocate the use of one notation or formalism
over another, the intended purpose must be stated and a strong argument
for the advantages for this particular purpose must be made.

>> Although I still find your equations a little hard to decipher,
>
> What is hard to decipher? d = exterior derivative, /\ = exterior product.
> I use \x to denote the direct product in the EMT, but if you are familiar
> with classical field theory you should know how T is calculated from the
> Lagrangian. I will leave out \x as I guess it doesn't help. I assume that
> one has been thru the field theory of the EM field. If not, this post
> won't make sense, but all physicists should be familiar with that.

I find your equations hard to decipher because you make many claims
without justification that are not obviously true, although they very well
might be. At least they are not obvious to me. I've taken a crack at
sorting some of your equations out in another branch of this thread, your
clarifications are welcome.

>>I think all of this is well known by now, even if it is usually expressed
>>in different language.
>
> Is it? The only place I have seen it expressed in any language is in
> Soper's "Classical Field Theory", and he left a lot to be done.

I don not know of where to find a thorough relativistic treatment of
continuum mechanics coupled to E&M. But some of this is worked out in
Chapter IV of Landau's Classical Field Theory (vol. 2) and Chapter XV of
his Hydrodynamics (vol. 6). Also, take a look at Chapter VIII of
Electrodynamics of Continuous Media by Landau and Lifshitz (vol. 8). There
the equations for non-relativistic MHD are worked out. You have a point
about possible pitfalls of using a Galilean theory for matter coupled to a
Lorentzian theory for the E&M field. However, the E&M equations can also
be reduced to a set of Galilean equations when the limit c -> oo is taken
into account. This amounts to dropping the terms corresponding to the
electric displacement current and the Faraday effect.

> The point I am making is that the theory is much more powerful in the
> field theoretic form. This is how things should be done in modern
> physics--it is how classical EM is done. The knowledge that A is the
> vector field for the field theory of EM, and that F = dA, and L = -
> (F|F)/2 is important and a powerful tool for problems in EM, as well as
> for integrating it with the rest of physics and for making progress.
>
> The same is true for continua. Everyone should know that the field theory
> of continua has 3 scalar fields, the Lagrangian coordinates, which come
> from the eqn. of continuity d*J = 0, and the Lagrangian L = - r + L_{em},
>
> where r = total energy density, and L_{em} = Lagrangian for any EM fields.

More powerful? That's an ambiguous statement that I discussed above.
I think it would be a safe guess that anyone working on a particular
theory knows what the relevant dynamical variables of that theory are.
But the choice of dynamical variables is not unique, you can also do a
change of variables. In E&M, all the equations can be expressed either in
terms of A or F. A is the variable with respect to which the variational
problem is constructed, but even then. I could define something like A' =
*A, or A' = A + dV, or A' = (A|.), or other possibly non-linear field
redefinitions. Just as with E&M, all hydrodynamical equations can be
expressed in terms of the current density j or as you do in terms of the
Lagrangian coordinates, or perhaps Eulerian coordinates or some other
field definition. Once again, you must state your goal and argue why your
choice of field is best.

Igor

Van Jacques

Oct28-04, 01:28 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>\n\nThe following are in response to some comments below\nand to my general experience in trying to talk to\nsomeone about this problem.\n========\nIs noone bothered that there is no field theory for\ncontinua--not even for cold dust moving in space?\nNot even cold dust in 1 dimension?\n\nThe problem is that people who are interested in fluids,\nMHD, plasma, and those who care about field theory are\nmutually exclusive. Its pointless to talk to one about\nthe other. Thus the field theory of fluids, MHD, plasma,\neven cold dust, gets left in the cold.\n\nGoldstein in "Classical Mechanics", and many other\ngreat physicists have attempted and failed to give\nsolutions to this problem, and only Soper ever gave\na solution. It was ignored, though he published it\nin his book, "Classical Field theory".\n============\nIgor Khavkine wrote:\n> I\'ve mentioned in a previous post that I have trouble interpreting\nyour\n> notation and conclusions. Let me try to interpret what I understand\nand\n> perhaps you can fill in some blanks.\n>\n> On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:\n>\n> > Now the point of the post. What about continua. What is the\nhomogenous\n> > eqn. which gives the field for continua? It is the eqn. of\ncontinuity. Let\n> > j = *J = n*u, where J = nu = matter current density, n = (rest)\nmass\n> > density, and u = 4-velocity.\n>\n> The first point is that I do not think it is a trivial fact that\n> current density is represented by a 3-form in 4D spacetime. Second, I\nam\n> confused by your use of the Hodge star. You seem to be claiming that\nJ\n> is a 1-form which is proportional to u, where u is a 4-velocity\nvector.\n> I guess the identification can be made with the metric, but this step\n> cannot be omitted.\n\nI am glad you want that kind of rigor, but the metric does indeed\nprovide\nand isomorphism between 4-vector fields and 1-forms. The way I was\ntaught was that this isomorphism is reflected in the fact that one\ncan use the metric to raise and lower indices. Since I didn\'t bother\nwith indices in this broad overall picture, I didn\'t think it\nnecessary to talk about the isomorphism between 1-forms and 4-vectors.\n\nAssuming there is no gravity so that space is flat, we have\n\nJ^a = n u^a ; u^a = dx^a/ds ; s = proper time ; a = 0,1,2,3\n\n= *d*J = *dj\n\nJ_a = g_ab J^b with g_ab = diag( -1,1,1,1)\n\nbut to readers of sci.physics.research, whom I assume are physics grad\n\nstudents or PhDs, I don\'t think this kind of thing is necessary.\nContinuing, the current density 3-form is j = *J, or\n\nj_abc = e_abcd J^d ; where e_abcd = 4D Levi-Cicita tensor = 4D volume\n\nelement d^4(x) = dV.\n\nThe eq. of continuity or conservation of matter becomes\n\n(with div = *d* from differential geometry), div(nu) = div(J)\n\n= *d*J = *dj = 0 ==> dj = 0 ; where d = A(grad) is the exterior\nderivative,\n\nand A = the antisymmetrizing operator, again from differential\ngeometry.\n\n\n> For my own benefit and that of anyone interested, let me sketch that\na\n> current density in an (k+1)-dimensional spacetime must indeed be an\n> k-form. Start off in familiar territory and assume that space and\ntime\n> are split.\n\nI don\'t like to split space and time until the end. The calculations\nare easier in 4D, and many relationships are clear in 4D that become\ninvisible in 3 + 1 notation. e.g. dF = 0 ==> F = dA is clearer and\neasier\nin 4D, but a mess in 3 + 1 notation.\n\n> Let n\' be the particle density (n\' integrated over a k-volume\n> of space gives a number, must be a k-form on space).\n\nAgain, lets use only Lorentz invariant quantities, like\nthe proper number density n in J = nu. Everything should be coordinate\nindependent, i.e. tensors (the term tensor includes Lorentz scalars.\nWhen ever one has a current _density, it is really a 3-form, as you\nsay.\nNote that we can then write n = sqrt[- (J|J)] ; which is obviously a\nscalar because of the scalar product (J|J = g(J,J).\n\n> Let j\' be the particle\n> current density. Given a (k-1)-dimensional hypersurface, the integral\nof\n> j\' over it also gives a number, the flux through the hypersurface per\n> unit time. Hence j\' must be a (k-1)-form on space. The continuity\n> equation takes the form (where V is a k-volume of space, and @V its\n> boundary)\n>\n> @/@t int_V n = int_(@V) j\n\nVery good. Yes, in 3 + 1 form, as it is called.\n\n> or equilvalently in integral form\n>\n> int_(V at t2) n\' - int_(V at t1) n\' = int_(t1 to t2) int_(@V) j\' dt\n> = int_(@Vx[t1,t2]) j\' /\\ dt .\n>\n> Now we have two k-forms on spacetime, n\' and j\' /\\ dt, let me denote\n> their sum by j = n\' + j\' /\\ dt. Looking at the above equation\ncarefully,\n> we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the\n> boundary of the spacetime volume Vx[t1,t2]. Now the continuity\nequation\n> can be rewritten as\n>\n> int_@(Vx[t1,t2]) j = 0 which implies int_(Vx[t1,t2]) dj = 0.\n>\n> And since I can build any spacetime volume out of little pieces that\n> look like Vx[t1,t2], the differential form of the continuity equation\n> is just dj = 0.\n>\n> The above discussion justifies two things. First that the current\n> density is given by a k-form j in a (k+1)-dimensional space time.\n> Integrated over a space-like k-volume, j gives the number of\nparticles\n> contained in it.\n\nExactly. You have made me very happy by this post. Thank you.\nYou are someone I would be delighted to talk with about anything at\nany time. If I can be of any help, feel free to ask me.\nYou can email me at vanjac12ATyahoo.com (replace AT with @).\n\nI would say that if V is a 3D volume, or in fact any 3D hypersurface in\n4D, then\n\nN = Int_V(j) = total # of particles (or equivalently the rest mass)\nin V at time x^0 = t,\nor if it is a sphere moving thru time, then it is the flux of fluid\nthru the sphere per unit time, as I think you said.\n\n> While integrated over a timelike k-hypersurface, j\n> gives the total number of particles that have passed through it.\nSecond\n> it justifies the simple dj = 0 form of the continuity equation in\n> space-time. I might have messed up with the sign of the second term\nin\n> j, perhaps it is dt /\\ j, but I think the general idea is fine.\n\nYes, you have it.\n\n> Now, about the relation between the velocity field u and the current\n> density j. They can\'t be simply related by a Hodge transformation,\nsince\n> one is a form and the other is a vector field. And how does the\n> application of the Hodge star follow from first principles anyway?\n\nThe 3-form dual to the 4-velocity (thought of as a 4-vector or a\n1-form,\nit doesn\'t matter), is the 3D volume element\n*u = d^3(x) = dx^1 /\\ dx^2 /\\ dx^3\northogonal to the time direction defined by the 4-velocity u = d/ds\n(recall that in a local comoving coord. system, u = @/@x^0 = @/@t,\nusing @ for the partial derivative (I usually write d for the partial\nderiv. if it won\'t create confusion).\n\n> > Then write the eqn. of continuity div(J) = 0\n> > as dj = 0. The exterior derivative of the 3-form j = 0, so that j\nis\n> > closed and therefore exact. Because of the isotropy of 3D space, we\nhave j\n> > is the exterior product of the exterior derivative of 3 scalar\nfields z^1,\n> > z^2, z^3.\n> >\n> > j = dz^1 /\\ dz^2 /\\ dz^3 = *J\n\nNever mind that for now. Its a sloppy way of justifying something\nthat thakes some experience to see.\n\nThe thing you need to do now is to do the problem in explicit detail\nin 1 space dimension, i.e. 2D spacetime. This allows one to get a grip\non everything.\n\nLet u^0 = dx^0/ds ; u^1 = dx^1/ds ; J = nu in 2D.\n\nThen the metric is g = diag(-1, 1), and the Hodge * operator is the\n2D Levi-Civita tensor with non-zero components\n\ne_01 = 1 = - e_10 ; e = dx^0 /\\ dx^1 is the 2D volume element\n\n(just as e is the 4D vol. element in 4D).\n\nj = *J = e(J) = e_ab J^a e^b ; where e^b = dx^b ; e_a = d/dx^a\n\nfrom differential geometry. Now,\n\ndj = 0 ==> j = dz = dz/dx^0 e^0 + dz/dx^1 e^1 = z_0 dx^0 + z_1 dx^1\n\nto get J^0 = ng = dz/dx^1 ; J^1 = ngv = - dz/dx^0\n\nWhere u^0 = g = 1/sqrt(1 - v^2) ; u^1 = gv ; v = dx^1/dx^0 =\n3-velocity.\n\nI usually write (x^0, x^1) --> (t,x) ; so v = dx/dt, etc.\n\nThen for cold dust, L = - n = - sqrt(j|j) = - sqrt(dz|dz)\nis the Lagrangian, and\nclassical field theory gives\n\nT = Lg - dz @ dL/d(dz) = - ng + dz@dz/n; @ denotes the tensor product.\n\nBut j = *J = n*u = dz, so dz@dz/n = n*u@*u (again, these come after\n\na lot of experience fooling around with this stuff).\nAlso, it turns out that\n\ng = - u @ u + *u @ *u ==> T = nuu ; div(T) = u[div(nu)] + n (u|grad)u\n\n= 0 + na = 0, ==> a = acceleration = 0 as we expect in this case.\n\nOne can then move on to 4D, perfect fluids, MHD, plasma, waves in\nall these media, the whole of the mechanics of continua can now\nbe formulated as a classical field theory, once we finally know what\nthe fields are.\n\nIn 4D there 3 scalar field z^i instead of just the one, z(x).\n\nI hope you follow this thru. I don\'t think you will regret it.\nIt is one of the most rewarding things I studied in a lifetime\nof doing physics.\n\nVan\n> I don\'t understand your last comment. How does "isotropy of 3D space"\n> lead to this particular form of j? By analogy with E&M, and by\nPoicare\'s\n> lemma, since dj = 0, then j = dL for some (k-1)-form A on spacetime.\n> What would this (k-1)-form dL represent?\n>\n> > If this is the way to do EM, then this is the way to do continuous\nmatter.\n> > Its clear to me that this is how to do both theories. Doing\nanything else\n> > leads to a\n> > mess and to errors. For example I do waves in perfect fluids, MHD,\nand\n> > plasma in my paper, including the energy and momentum of the waves,\nand\n> > the eqns. obeyed by the waves as they propagate, as well as the\nusual\n> > dispersion and polarization relations. Everything falls out easily\nif one\n> > start from this, the correct framework.\n>\n> As far as I know, people have been doing hydrodynamics before\n> differential geometry in its current for was formulated. So notation\nand\n> formalism are definitely not an obstacle for theoretical physics. It\nmay\n> be true that using the notation you suggest it is easier to get all\nthe\n> equations of motion and such, but it is certainly not the only way.\nThe\n> most probably reason for this formalism not being used in\nhydrodynamics\n> is that people who are doing the latter are usually not familiar with\n> the latter.\n>\n> > As I said, Soper published some of this in his book, but no one\nseems to\n> > have noticed.\n>\n> Most probably so. And they probably never will unless someone points\nit\n> out along with definite advantages to introducing a new formalism in\n> their work.\n>\n> Igor\nSee the beginning for my response.\nI have written a paper with solution of this fundamental\nproblem.\n\nVan\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>The following are in response to some comments below
and to my general experience in trying to talk to
someone about this problem.
========
Is noone bothered that there is no field theory for
continua--not even for cold dust moving in space?
Not even cold dust in 1 dimension?

The problem is that people who are interested in fluids,
MHD, plasma, and those who care about field theory are
mutually exclusive. Its pointless to talk to one about
the other. Thus the field theory of fluids, MHD, plasma,
even cold dust, gets left in the cold.

Goldstein in "Classical Mechanics", and many other
great physicists have attempted and failed to give
solutions to this problem, and only Soper ever gave
a solution. It was ignored, though he published it
in his book, "Classical Field theory".
============
Igor Khavkine wrote:
> I've mentioned in a previous post that I have trouble interpreting
your
> notation and conclusions. Let me try to interpret what I understand
and
> perhaps you can fill in some blanks.
>
> On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:
>
> > Now the point of the post. What about continua. What is the
homogenous
> > eqn. which gives the field for continua? It is the eqn. of
continuity. Let
> > j = *J = n*u, where J = \nu = matter current density, n = (rest)
mass
> > density, and u = 4-velocity.
>
> The first point is that I do not think it is a trivial fact that
> current density is represented by a 3-form in 4D spacetime. Second, I
am
> confused by your use of the Hodge star. You seem to be claiming that
J
> is a 1-form which is proportional to u, where u is a 4-velocity
vector.
> I guess the identification can be made with the metric, but this step
> cannot be omitted.

I am glad you want that kind of rigor, but the metric does indeed
provide
and isomorphism between 4-vector fields and 1-forms. The way I was
taught was that this isomorphism is reflected in the fact that one
can use the metric to raise and lower indices. Since I didn't bother
with indices in this broad overall picture, I didn't think it
necessary to talk about the isomorphism between 1-forms and 4-vectors.

Assuming there is no gravity so that space is flat, we have

J^a = n u^a ; u^a = dx^a/ds ; s =[/itex] proper time ; a = 0,1,2,3= *d*J = *djJ_a = g_{ab} J^b with g_{ab} = diag( -1,1,1,1)

but to readers of sci.physics.research, whom I assume are physics grad

students or PhDs, I don't think this kind of thing is necessary.
Continuing, the current density 3-form is j = *J, or

j_{abc} = e_{abcd} J^d ; where e_{abcd} = 4D Levi-Cicita tensor = 4D volume

element d^4(x) = dV.

The eq. of continuity or conservation of matter becomes

(with div = *d* from differential geometry), div(\nu) = div(J)= *d*J = *dj = ==> dj = ; where d = A(grad) is the exterior
derivative,

and A = the antisymmetrizing operator, again from differential
geometry.

> For my own benefit and that of anyone interested, let me sketch that
a
> current density in an (k+1)-dimensional spacetime must indeed be an
> k-form. Start off in familiar territory and assume that space and
time
> are split.

I don't like to split space and time until the end. The calculations
are easier in 4D, and many relationships are clear in 4D that become
invisible in 3 + 1 notation. e.g. dF = ==> F = dA is clearer and
easier
in 4D, but a mess in 3 + 1 notation.

> Let n' be the particle density (n' integrated over a k-volume
> of space gives a number, must be a k-form on space).

Again, lets use only Lorentz invariant quantities, like
the proper number density n in J = \nu. Everything should be coordinate
independent, i.e. tensors (the term tensor includes Lorentz scalars.
When ever one has a current _density, it is really a 3-form, as you
say.
Note that we can then write n = \sqrt[- (J|J)] ; which is obviously a
scalar because of the scalar product (J|J = g(J,J).

> Let j' be the particle
> current density. Given a (k-1)-dimensional hypersurface, the integral
of
> j' over it also gives a number, the flux through the hypersurface per
> unit time. Hence j' must be a (k-1)-form on space. The continuity
> equation takes the form (where V is a k-volume of space, and @V its
> boundary)
>
> @/@t \int_V n = \int_(@V) j

Very good. Yes, in 3 + 1 form, as it is called.

> or equilvalently in integral form
>
> \int_(V at t2) n' - \int_(V at t1) n' = \int_(t1 to t2) \int_(@V) j' dt
> = \int_(@Vx[t1,t2]) j' /\ dt .
>
> Now we have two k-forms on spacetime, n' and j' /\ dt, let me denote
> their sum by j = n' + j' /\ dt. Looking at the above equation
carefully,
> we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the
> boundary of the spacetime volume Vx[t1,t2]. Now the continuity
equation
> can be rewritten as
>
> \int_@(Vx[t1,t2]) j = which implies \int_(Vx[t1,t2]) dj = .
>
> And since I can build any spacetime volume out of little pieces that
> look like Vx[t1,t2], the differential form of the continuity equation
> is just dj = .
>
> The above discussion justifies two things. First that the current
> density is given by a k-form j in a (k+1)-dimensional space time.
> Integrated over a space-like k-volume, j gives the number of
particles
> contained in it.

Exactly. You have made me very happy by this post. Thank you.
You are someone I would be delighted to talk with about anything at
any time. If I can be of any help, feel free to ask me.
You can email me at vanjac12ATyahoo.com (replace AT with @).

I would say that if V is a 3D volume, or in fact any 3D hypersurface in
4D, then

N = \Int_V(j) = total # of particles (or equivalently the rest mass)
in V at time x^0 = t,
or if it is a sphere moving thru time, then it is the flux of fluid
thru the sphere per unit time, as I think you said.

> While integrated over a timelike k-hypersurface, j
> gives the total number of particles that have passed through it.
Second
> it justifies the simple dj = form of the continuity equation in
> space-time. I might have messed up with the sign of the second term
in
> j, perhaps it is dt /\ j, but I think the general idea is fine.

Yes, you have it.

> Now, about the relation between the velocity field u and the current
> density j. They can't be simply related by a Hodge transformation,
since
> one is a form and the other is a vector field. And how does the
> application of the Hodge star follow from first principles anyway?

The 3-form dual to the 4-velocity (thought of as a 4-vector or a
1-form,
it doesn't matter), is the 3D volume element
*u = d^3(x) = dx^1 /\ dx^2 /\ dx^3
orthogonal to the time direction defined by the 4-velocity u = d/ds
(recall that in a local comoving coord. system, u = @/@x^0 = @/@t,
using @ for the partial derivative (I usually write d for the partial
deriv. if it won't create confusion).

> > Then write the eqn. of continuity div(J) =
> > as dj = . The exterior derivative of the 3-form j = 0, so that j
is
> > closed and therefore exact. Because of the isotropy of 3D space, we
have j
> > is the exterior product of the exterior derivative of 3 scalar
fields z^1,
> > z^2, z^3.
> >
> > j = dz^1 /\ dz^2 /\ dz^3 = *J

Never mind that for now. Its a sloppy way of justifying something
that thakes some experience to see.

The thing you need to do now is to do the problem in explicit detail
in 1 space dimension, i.e. 2D spacetime. This allows one to get a grip
on everything.

Let u^0 = dx^0/ds ; u^1 = dx^1/ds ; J = \nu in 2D.

Then the metric is g = diag(-1, 1), and the Hodge * operator is the
2D Levi-Civita tensor with non-zero components

e_{01} = 1 = - e_{10} ; e = dx^0 /\ dx^1 is the 2D volume element

(just as e is the 4D vol. element in 4D).

j = *J = e(J) = e_{ab} J^a e^b ; where e^b = dx^b ; e_a = d/dx^a

from differential geometry. Now,

dj = ==> j = dz = dz/dx^0 e^0 + dz/dx^1 e^1 = z_0 dx^0 + z_1 dx^1

to get J^0 = ng = dz/dx^1 ; J^1 = ngv = - dz/dx^0

Where u^0 = g = 1/\sqrt(1 - v^2) ; u^1 = gv ; v = dx^1/dx^0 =
3-velocity.

I usually write (x^0, x^1) --> (t,x) ; so v = dx/dt, etc.

Then for cold dust, L = - n = - \sqrt(j|j) = - \sqrt(dz|dz)
is the Lagrangian, and
classical field theory gives

T = Lg - dz @ dL/d(dz) = - ng + dz@dz/n; @ denotes the tensor product.

But j = *J = n*u = dz, so dz@dz/n = n*u@*u (again, these come after

a lot of experience fooling around with this stuff).
Also, it turns out that

g = - u @ u + *u @ *u ==> T = nuu ; div(T) = u[div(\nu)] + n (u|grad)u

= + na = 0, ==> a = acceleration = as we expect in this case.

One can then move on to 4D, perfect fluids, MHD, plasma, waves in
all these media, the whole of the mechanics of continua can now
be formulated as a classical field theory, once we finally know what
the fields are.

In 4D there 3 scalar field z^i instead of just the one, z(x).

I hope you follow this thru. I don't think you will regret it.
It is one of the most rewarding things I studied in a lifetime
of doing physics.

Van
> I don't understand your last comment. How does "isotropy of 3D space"
> lead to this particular form of j? By analogy with E&M, and by
Poicare's
> lemma, since [itex]dj = 0, then j = dL for some (k-1)-form A on spacetime.
> What would this (k-1)-form dL represent?
>
> > If this is the way to do EM, then this is the way to do continuous
matter.
> > Its clear to me that this is how to do both theories. Doing
anything else
> > leads to a
> > mess and to errors. For example I do waves in perfect fluids, MHD,
and
> > plasma in my paper, including the energy and momentum of the waves,
and
> > the eqns. obeyed by the waves as they propagate, as well as the
usual
> > dispersion and polarization relations. Everything falls out easily
if one
> > start from this, the correct framework.
>
> As far as I know, people have been doing hydrodynamics before
> differential geometry in its current for was formulated. So notation
and
> formalism are definitely not an obstacle for theoretical physics. It
may
> be true that using the notation you suggest it is easier to get all
the
> equations of motion and such, but it is certainly not the only way.
The
> most probably reason for this formalism not being used in
hydrodynamics
> is that people who are doing the latter are usually not familiar with
> the latter.
>
> > As I said, Soper published some of this in his book, but no one
seems to
> > have noticed.
>
> Most probably so. And they probably never will unless someone points
it
> out along with definite advantages to introducing a new formalism in
> their work.
>
> Igor
See the beginning for my response.
I have written a paper with solution of this fundamental
problem.

Van

Arnold Neumaier

Oct28-04, 01:28 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>\n\n\n\nVan Jacques wrote:\n\n> I don\'t want to be critical of other\'s work, but\n> I believe there is room for\n> improvement using the ideas I present here.\n>\n> I will email them, but experience has taught me that by\n> no longer working at a university I have lost my voice.\n> I believe these ideas are important.\n\nThe secret is that most people like to answer questions that\nfall into their field of expertise, if it does not take too\nmuch effort to reply. But few like to listen to half-baked\n(or even fully baked but only outlined) ideas;\ntoo many such offers come from cranks. The devil is always\nin the details; and if you can\'t provide them it is likely\nthey\'ll think it is because it does not work or does not offer\nany advantage.\n\nSo the right approach is to ask them for information about\nwhats known in the direction you want to go, rather than\nproposing a revolutionary way of doing it correctly.\n\nIf you really can do it better than others, and you don\'t find\nprior relevant work in the literature, work it out yourself and\nshow with a nontrivial application that you can do something more\neffciently than tradition. Then submit it to a respectable\njournal, and people are more likely to listen.\n\nAnd even if your work is good but not mainstream, it may take\npersistence to publicize it properly; publishing is not enough.\n\n\nArnold Neumaier\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>Van Jacques wrote:

> I don't want to be critical of other's work, but
> I believe there is room for
> improvement using the ideas I present here.
>
> I will email them, but experience has taught me that by
> no longer working at a university I have lost my voice.
> I believe these ideas are important.

The secret is that most people like to answer questions that
fall into their field of expertise, if it does not take too
much effort to reply. But few like to listen to half-baked
(or even fully baked but only outlined) ideas;
too many such offers come from cranks. The devil is always
in the details; and if you can't provide them it is likely
they'll think it is because it does not work or does not offer
any advantage.

So the right approach is to ask them for information about
whats known in the direction you want to go, rather than
proposing a revolutionary way of doing it correctly.

If you really can do it better than others, and you don't find
prior relevant work in the literature, work it out yourself and
show with a nontrivial application that you can do something more
effciently than tradition. Then submit it to a respectable
journal, and people are more likely to listen.

And even if your work is good but not mainstream, it may take
persistence to publicize it properly; publishing is not enough.

Arnold Neumaier

Arnold Neumaier

Oct29-04, 02:07 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>\n\nVan Jacques wrote:\n\n> The problem is that people who are interested in fluids,\n> MHD, plasma, and those who care about field theory are\n> mutually exclusive. Its pointless to talk to one about\n> the other. Thus the field theory of fluids, MHD, plasma,\n> even cold dust, gets left in the cold.\n\nIf you want to change that, you must play the translator\nand messenger between them...\n\nWrite an exposition with one chapter each which contains the\nbackground each party needs in order to understand the other side,\nstarting from what you know they know already, and then a third\nchapter explaining what can be gained from putting both sides together.\n\n\nArnold Neumaier\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>Van Jacques wrote:

> The problem is that people who are interested in fluids,
> MHD, plasma, and those who care about field theory are
> mutually exclusive. Its pointless to talk to one about
> the other. Thus the field theory of fluids, MHD, plasma,
> even cold dust, gets left in the cold.

If you want to change that, you must play the translator
and messenger between them...

Write an exposition with one chapter each which contains the
background each party needs in order to understand the other side,
starting from what you know they know already, and then a third
chapter explaining what can be gained from putting both sides together.

Arnold Neumaier

Van Jacques

Oct30-04, 08:10 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\nIgor Khavkine wrote:\n> On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:\n>\n\nFIELD THEORY FOR A PERFECT FLUID IN 2D SPACETIME\n\nConsider a CMCS (T,z) = (z^0,z^1)\nin 2D spacetime. Let the fluid have pressure p given by\nits internal or thermal energy q(n). The 1st law of thermo is\n\nT_e dS = dq + p d(1/n) ; Assume the flow is isentropic; S = const.\n\ndS = 0 ==> p = n^2 dq/dn ; dp = n df or dr = f dn\n\nwhere r = n(1 + q) = total proper energy density\nT_e = temperature ; S = entropy ;\nf = 1 + q + p/n = relativistic enthalpy.\n\nDefine c = fu. We also have L = - r. The Euler Lagrange can\nbe dealt with by absorbing n_o into z, so that j = dz,\nwhich follows from dj = 0.\n\nThe Euler-Lagrange (EL) eq. is\n\ndiv(dL/d(dz)) = 0 ; - dL/d(dz) = dr/d(dz) = f dn/d(dz)\n\n= f d sqrt(dz|dz)/d(dz) = f dz/n = *(fu) = *c ; where c = fu;\n\nThe EL eq. is\ndiv(*c) = *d**(fu) = *d(fu) ==> d(fu) = dc = 0\n\n==> c = dT = fu ; This suggests that we should choose\n\nz^0 = T, e^0 = dz^0 = dT , u = dT/f ;and\nz^1 = z, e^1 = dz = j = n*u , *u = dz/n .\n\nSo in the CMCS (T,z) the e^i are not unit vectors, though\nthey are orthogonal (in 2D)., since\ng^01 = g(e^0,e^1) = (dT|dz) = nf *u(u) = 0.\n\ng^00 = (dT|dT) = - f^2 ; g^11 = (dz|dz) = n^2 ==>\ndet(g) = -1/(nf)^2 , and\n\ng = - uu + *u*u = - dT dT/f^2 + dz dz/n^2 in a CMCS.\n\nThe energy-momentum tensor (EMT) is (the EMT T not to be confused with\nz^0 = T)\n\nT = Lg - dz dL/d(dz) = - rg + fdz dz/n = - rg + nf *u *u\n\nnf = r + p, so T = ruu + p*u*u = nfuu + pg.\n\nWe have 2 equivalent eqns. from which to get the eq. of motion;\n\nThe EL eq, d(fu) = df /\\ u + f du = 0; take scalar product with u;\n\nP(df) + fa = 0 ; where P = g + uu = *u*u in 2D is the spatial\n\nprojection operator, and a = du(u) = acceleration.\n\nUsing the 1st law of thermo above gives\n\na = - P(df/f) ; or nfa + P(dp) = 0\n\nwhich is just the Euler eqn. of motion for a perfect fluid.\n\nThe EMT ==> div(T) = div(nfuu) + dp = nu df(u) + nfa + dp\n\n(u|div(T)) = dp(u) - n df(u) = T_e dS(u) = 0 ; this is conservation\n\nof energy for adiabiatic flow with dS(u) = 0, i.e., entropy\nis const. along streamlines. The spatial part of div(T) = 0 is\n\nP(div(T) = nfa + P(dp) = 0 ; again, the Euler eq. of motion\nfor a perfect fluid.\n\nVan\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>Igor Khavkine wrote:
> On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:
>

FIELD THEORY FOR A PERFECT FLUID IN 2D SPACETIME

Consider a CMCS (T,z) = (z^0,z^1)
in 2D spacetime. Let the fluid have pressure p given by
its internal or thermal energy q(n). The 1st law of thermo is

T_e dS = dq + p d(1/n) ;[/itex] Assume the flow is isentropic; S = const.

dS = ==> p = n^2 dq/dn ; dp = n df or dr = f dn

where r = n(1 + q) = total proper energy density
T_e = temperature ; S = entropy ;
f = 1 + q + p/n = relativistic enthalpy.

Define c = fu. We also have L = - r. The Euler Lagrange can
be dealt with by absorbing n_o into z, so that j = dz,
which follows from dj = .

The Euler-Lagrange (EL) eq. is

div(dL/d(dz)) =; - dL/d(dz) = dr/d(dz) = f dn/d(dz)= f d \sqrt(dz|dz)/d(dz) = f dz/n = *(fu) = *c ; where c = fu;

The EL eq. is
div(*c) = *d**(fu) = *d(fu) ==> d(fu) = dc =

==> c = dT = fu ; This suggests that we should choose

z^0 = T, e^0 = dz^0 = dT , u = dT/f ;and
z^1 = z, e^1 = dz = j = n*u , *u = dz/n .

So in the CMCS (T,z) the e^i are not unit vectors, though
they are orthogonal (in 2D)., since
g^{01} = g(e^0,e^1) = (dT|dz) = nf *u(u) = .

g^{00} = (dT|dT) = - f^2 ; g^{11} = (dz|dz) = n^2 ==>
det(g) = -1/(nf)^2 , and

g = - uu + *u*u = - dT dT/f^2 + dz dz/n^2 in a CMCS.

The energy-momentum tensor (EMT) is (the EMT T not to be confused with
z^0 = T)T = Lg - dz dL/d(dz) = - rg + fdz dz/n = - rg + nf *u *unf = r + p, so T = ruu + p*u*u = nfuu + pg.

We have 2 equivalent eqns. from which to get the eq. of motion;

The EL eq, d(fu) = df /\ u + f du = 0; take scalar product with u;

P(df) + fa = ; where P = g + uu = *u*u in 2D is the spatial

projection operator, and a = du(u) = acceleration.

Using the 1st law of thermo above gives

a = - P(df/f) ; or nfa [itex]+ P(dp) =

which is just the Euler eqn. of motion for a perfect fluid.

The EMT ==> div(T) = div(nfuu) + dp = \nu df(u) + nfa + dp(u|div(T)) = dp(u) - n df(u) = T_e dS(u) = ; this is conservation

of energy for adiabiatic flow with dS(u) = 0, i.e., entropy
is const. along streamlines. The spatial part of div(T) = is

P(div(T) = nfa + P(dp) = ; again, the Euler eq. of motion
for a perfect fluid.

Van

Van Jacques

Nov2-04, 12:38 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>\n\n\nVan Jacques wrote:\n> Igor Khavkine wrote:\n> > On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:\n\n\nRELATIVISTIC FIELD THEORY FOR A PERFECT FLUID\n\n>From the 1st law of thermo for isentropic flow,\n\nde + p d(1/n) = de - p dn/n^2 = 0 ;\n\ne = internal (thermal) energy\nr = n(1 + e)\nf = 1 + e + p/n\n\nWe find that dp = n df ; dr = f dn, which are useful later.\n\nj = dz^1 /\\ dz^2 /\\ dz^3 = *J = n*u ;\n\nThe comps of J in a Cartesian CS x are\n\nJ^0 = nu^0 = ng = d(z^1,z^2,z^3)/d(x^1,x^2,x^3) = Z_123\n\nwhere g = 1/sqrt(1 - v^2).\n\nIn general, define Z_ijk = d(z^1,z^2,z^3)/d(x^i,x^j,x^k). Then\n\nJ^1 = - Z_023 ; J^2 = Z_013 ; J^3 = - Z_012 ; so\n\nJ^i = nu^i = (-1)^i Z_0 \\i\\ 3; where \\i\\ denotes that i is omitted.\n\nThe components of the 3-velocity are v^i = J^i/J^0 .\n\nThe Lagrangian is L = - r = - n(1 + e), where\ne(n) depends on n only for a perfect fluid.\n\nn^2 = (j|j( = (dz^1/\\dz^2/\\dz^3|dz^1/\\dz^2/\\dz^3)\n\n= det(dz^i|dz^j) = det G ; G^ij = (dz^i|dz^j)\n\nIts useful to note the G is just the spatial part of\nthe metric in the CMCS z. I will leave the details of\nthe calculations that follow to readers.\n\nThe energy-momentum tensor (EMT) T is\n\nT = Lg - DL ; where DL = dz^i (x) dL/d(dz^i) ; (x) = tensor product\n\nDL = - Dr = - f Dn = - nf (g + uu) = - nf P ; since Dn = nP\n\nwhere P = g + uu is the spatial projection operator (P(u) = 0).\n(this is a central, but long calcuation--lots of algebra).\n\nT = ruu + pP = diag(r,p,p,p) ; which is as it should be, or\n\nT = nfuu + pg (useful for calculations).\n\nT(u,u) = total energy density = r\nPTP = pP = (isotropic) pressure tensor\n\ndiv(T) = nu df(u) + nfa + dp = 0; (using div(nu) = 0) ==>\n\n(u|div(T)) = dp(u) - n df(u) = 0; cons. of energy, and\n\nP(div(T)) = nfa + P(dp) = 0; the usual Euler eqn. for a perfect fluid.\n\nOne can also do more with vorticity and circulation, which\nI will post in the future.\n\nNext I will do relativistic MHD, a great improvement on non-rel MHD.\nVan\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>Van Jacques wrote:
> Igor Khavkine wrote:
> > On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:

RELATIVISTIC FIELD THEORY FOR A PERFECT FLUID

>From the 1st law of thermo for isentropic flow,

de + p d(1/n) = de - p dn/n^2 =[/itex] ;

e = internal (thermal) energy
r = n(1 + e)f = 1 + e + p/n

We find that dp = n df ; dr = f dn, which are useful later.

j = dz^1 /\ dz^2 /\ dz^3 = *J = n*u ;

The comps of J in a Cartesian CS x are

J^0 = \nu^0 = ng = d(z^1,z^2,z^3)/d(x^1,x^2,x^3) = Z_{123}

where g = 1/\sqrt(1 - v^2).

In general, define Z_{ijk} = d(z^1,z^2,z^3)/d(x^i,x^j,x^k). Then

J^1 = - Z_{023} ; J^2 = Z_{013} ; J^3 = - Z_{012} ; so

J^i = \nu^i = (-1)^i Z_0 \i\ 3; where \i\ denotes that i is omitted.

The components of the 3-velocity are v^i = J^i/J^0 .

The Lagrangian is L = - r = - n(1 + e), where
e(n) depends on n only for a perfect fluid.

n^2 = (j|j( = (dz^1/\dz^2/\dz^3|dz^1/\dz^2/\dz^3)= det(dz^i|dz^j) = det G ; G^{ij} = (dz^i|dz^j)

Its useful to note the G is just the spatial part of
the metric in the CMCS z. I will leave the details of
the calculations that follow to readers.

The energy-momentum tensor (EMT) T is

T = Lg - DL ; where DL = dz^i (x) dL/d(dz^i) ; (x) = tensor product

DL = - Dr = - f Dn = - nf (g + uu) = - nf P ; since [itex]Dn = nP

where P = g + uu is the spatial projection operator (P(u) = 0).
(this is a central, but long calcuation--lots of algebra).

T = ruu + pP = diag(r,p,p,p) ; which is as it should be, or

T = nfuu + pg (useful for calculations).

T(u,u) = total energy density = r
PTP = pP = (isotropic) pressure tensor

div(T) = \nu df(u) + nfa + dp = 0; (using div(\nu) = 0) ==>

(u|div(T)) = dp(u) - n df(u) = 0; cons. of energy, and

P(div(T)) = nfa + P(dp) = 0; the usual Euler eqn. for a perfect fluid.

One can also do more with vorticity and circulation, which
I will post in the future.

Next I will do relativistic MHD, a great improvement on non-rel MHD.
Van

Van Jacques

Nov3-04, 09:44 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>Van Jacques wrote:\n> Arnold Neumaier wrote:\n> > Van Jacques wrote:\n>What is the field for the field theory of cold dust in spacetime?\n\nSurely one must be able to answer this question before one\ncan say anything about field theoretic methods for continua.\nI have posted this many times, no one seems to care, and\npoints me to articles with calculations by people who can\'t\nanswer this basic question.\n\n********Classical field theory********\nDoes everyone agree with the following?\n\nThe action S = integral over 4D spacetime of the Lagrangian\ndensity L. L is a function of the field(s) and their 1st\npartial derivatives.\n\nL(z, dz; x) ; S = Int(L d^4(x)) ; d^4(x) = dx^0 /\\ dx^1 /\\ dx^2 /\\\ndx^3\n\nis the 4D volume element.\nx = (x^0,x^1,x^2,x^3) are the coords. on 4D spacetime.\n(We could say that S = Int(*L) ; * = Hodge * operator).\n\nConsider the Klien-Gordon eqn. with 1 scalar field z(x). Then\n\nL = (dz|dz)/2 ; where dz = grad(z), d = exterior derivative\n\nFor electromagnetism, dF = 0 ==> F = dA , and the 4-vector potential\nA (a 1-form) is the field for the field theory. The Lagrangian is\n\nL = - (dA|dA)/2 + (J|A)\n\nThe canonical energy momentum tensor (EMT) is\n\nT = gL - dz dL/d(dz) which gives the energy and momentum\n\ndensities and fluxes. div(T) = 0 gives the eqns. of conservation of\nenergy-momentum, which are the eqns. of motion.\nEveryone knows this, I am just establishing notation and\nthe framework.\n===============\n*********\nWhat is the field for the field theory of cold dust in spacetime?\nWhat is the Lagrangian? (a relativistic theory, not L = n v^2/2 -\nV(x)).\n*********\nVan Jacques\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>Van Jacques wrote:
> Arnold Neumaier wrote:
> > Van Jacques wrote:
>What is the field for the field theory of cold dust in spacetime?

Surely one must be able to answer this question before one
can say anything about field theoretic methods for continua.
I have posted this many times, no one seems to care, and
points me to articles with calculations by people who can't
answer this basic question.

********Classical field theory********
Does everyone agree with the following?

The action S = integral over 4D spacetime of the Lagrangian
density L. L is a function of the field(s) and their 1st
partial derivatives.

L(z, dz; x) ; S = \Int(L d^4(x)) ; d^4(x) = dx^0 /\ dx^1 /\ dx^2 /\dx^3

is the 4D volume element.
x = (x^0,x^1,x^2,x^3) are the coords. on 4D spacetime.
(We could say that S = \Int(*L) ; * = Hodge * operator).

Consider the Klien-Gordon eqn. with 1 scalar field z(x). Then

L = (dz|dz)/2 ;[/itex] where dz = grad(z), d = exterior derivative

For electromagnetism, dF = ==> F = dA , and the 4-vector potential
A (a 1-form) is the field for the field theory. The Lagrangian is

[itex]L = - (dA|dA)/2 + (J|A)

The canonical energy momentum tensor (EMT) is

T = gL - dz dL/d(dz) which gives the energy and momentum

densities and fluxes. div(T) = gives the eqns. of conservation of
energy-momentum, which are the eqns. of motion.
Everyone knows this, I am just establishing notation and
the framework.
===============
*********
What is the field for the field theory of cold dust in spacetime?
What is the Lagrangian? (a relativistic theory, not L = n v^2/2 -V(x)).*********
Van Jacques

Van Jacques

Nov3-04, 09:54 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>Van Jacques wrote:\n> > On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:\n>\n\nTHE CLASSICAL FIELD THEORY FOR (CONTINUOUS) MATTER IN SPACETIME\n\nFirst consider the simplest problem;\ncold (pressure = temp = 0) matter in flat 2D\nspace-time (1 spatial dimension).\nChoose the Cartesian coord. system (CS)\n(x^0,x^1) = (t,x) so that the matter is at rest.\nThe vector field u = d/ds defines the proper time s = s(t,x)\n= the time as seen by the particle at (t,x).\nFor matter at rest s = t and u = d/dt = e_0 = unit vector in the\ntime direction. The integral curves of u are the streamlines\nor worldlines of the fluid particles.\nThe metric in the (t,x) CS is g = diag(-1,1). The coordinate basis for\nthe tangent space is e_i = d/dx^i and the\ndual basis for the cotangent space is e^i = dx^i (i = 0,1 in 2D).\n(d denotes the exterior derivative everywhere\nexcept in e_i where d is the partial derivative.)\n\nThe initial value problem. Denote the\ninitial position of a fluid particle as z = x(t = 0).\nz = the Lagrangian coordinate of the particle\n= its initial position = its "label"--i.e., each fluid particle\nis labeled by its initial position. This label moves with\nthe particle and forms the spatial part of a comoving\ncoordinate system (CMCS). By the definition of z = z(t,x),\ndz = e^1 = dx for a fluid at rest.\ndz(u) = e^1(e_0) = 0 ==> z is const. along streamlines.\n\nNow that the floor has been laid down we can start to move\naround. (Pretty soon we will be dancing!)\n\nThe matter current vector J = nu = ne_0 = (n,0), where\nn = proper number density = number density measured in a CS\nmoving with the fluid.\n\nThe initial spatial hyper-surface V is defined by f(t,x) = t = 0.\nz is the coordinate on V. In 2D V is just a line with coordinate z. z(t\n= 0,x) = x. When the fluid is at rest,\nz = x = constant for all t.\n\nThe eq. of continuity is\n\ndiv(J) = *d*J = *dj = 0 ==> dj = 0 ==> j = dw\n\nj is a 1-form and w = w(t,x) is a 0-form = scalar fcn.\n* is the Hodge * operator, e = *1 = dt /\\ dx = antisymmetric 2D\nLevi-Civita tensor = the volume element in 2D spacetime.\n\nj = *J = e(J) = n*u = e_ik J^i e^k = dw = dw/dt dt + dw/dx dx\n= w_0 e^0 + w_1 e^1; j(u) = j_0 = dw/dt = n*u(u) = ne(u,u) = 0\nsince e is antisymmetric. ==>\nj = j_1 e^1 = w_1 dx = n_o dz ; (z = x for a fluid at rest.)\n\nFor now, n_o is just some function. It will turn out to\nbe the density on z-space V. Note\n\n0 = dj = dn_o /\\ dz = dn_o/dt dt /\\ dz ==> dn_o/dt = 0,\nso n_o = n_o(z).\n\nJ = nu = ne_0 = *j = n_o *dz = n_o e_0 ; *dz = nu/n_o\n\nFor a fluid at rest n = n_o(z), *dz = u ; *u = dz = dx,\nas one would expect--the fluid just sits there with constant density.\n\nThe Lagrangian is L = - n = - sqrt(j|j) = - n_o sqrt(dz|dz).\nThe canonical energy-momentum tensor (EMT) is\n\nT = Lg - dz @ dL/d(dz) ; @ denotes the tensor product.\n\nT = Lg + n dz@dz ; g = - u@u + dz@dz ; u = - dt as a 1-form;\n\nT = n(u@u - dz@dz) + n dz@dz = nu@u\n\ndiv(T) = 0 ==> na = 0 ; where a = du(u) = acceleration = 0\n\nas it must be, and div(nu) = 0; we get back the eq. of continuity.\n\nNext, introduce pressure in 2D spacetime. Look at the\ntransformation of coordinates from Cartesian (t,x) to\ncomoving (s,z). Now things become more interesting. The above\nwas mainly to establish notation and techniques. Now some physics.\nVan Jacques\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>Van Jacques wrote:
> > On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:
>

THE CLASSICAL FIELD THEORY FOR (CONTINUOUS) MATTER IN SPACETIME

First consider the simplest problem;
cold (pressure = temp = 0) matter in flat 2D
space-time (1 spatial dimension).
Choose the Cartesian coord. system (CS)
(x^0,x^1) = (t,x) so that the matter is at rest.
The vector field u = d/ds defines the proper time s = s(t,x)
= the time as seen by the particle at (t,x).
For matter at rest s = t and u = d/dt = e_0 = unit vector in the
time direction. The integral curves of u are the streamlines
or worldlines of the fluid particles.
The metric in the (t,x) CS is g = diag(-1,1). The coordinate basis for
the tangent space is e_i = d/dx^i and the
dual basis for the cotangent space is e^i = dx^i (i = 0,1 in 2D).
(d denotes the exterior derivative everywhere
except in e_i where d is the partial derivative.)

The initial value problem. Denote the
initial position of a fluid particle as z = x(t = 0).
z = the Lagrangian coordinate of the particle
= its initial position = its "label"--i.e., each fluid particle
is labeled by its initial position. This label moves with
the particle and forms the spatial part of a comoving
coordinate system (CMCS). By the definition of z = z(t,x),dz = e^1 = dx for a fluid at rest.
dz(u) = e^1(e_0) = ==> z is const. along streamlines.

Now that the floor has been laid down we can start to move
around. (Pretty soon we will be dancing!)

The matter current vector J = \nu = ne_0 = (n,0), where
n = proper number density = number density measured in a CS
moving with the fluid.

The initial spatial hyper-surface V is defined by f(t,x) = t = .
z is the coordinate on V. In 2D V is just a line with coordinate z. z(t= 0,x) = x. When the fluid is at rest,
z = x = constant for all t.

The eq. of continuity is

div(J) = *d*J = *dj =[/itex] ==> dj = ==> j = dw

j is a 1-form and w = w(t,x) is a 0-form = scalar fcn.
* is the Hodge * operator, e = *1 = dt /\ dx = antisymmetric 2D
Levi-Civita tensor = the volume element in 2D spacetime.

j = *J = e(J) = n*u = e_{ik} J^i e^k = dw = dw/dt dt + dw/dx dx= w_0 e^0 + w_1 e^1; j(u) = j_0 = dw/dt = n*u(u) = ne(u,u) =
since e is antisymmetric. ==>
j = j_1 e^1 = w_1 dx = n_o dz ; (z = x for a fluid at rest.)

For now, n_o is just some function. It will turn out to
be the density on z-space V. Note

= dj = dn_o /\ dz = dn_o/dt dt /\ dz ==> dn_o/dt = 0,
so n_o = n_o(z).

[itex]J = \nu = ne_0 = *j = n_o *dz = n_o e_0 ; *dz = \nu/n_o

For a fluid at rest n = n_o(z), *dz = u ; *u = dz = dx,
as one would expect--the fluid just sits there with constant density.

The Lagrangian is L = - n = - \sqrt(j|j) = - n_o \sqrt(dz|dz).
The canonical energy-momentum tensor (EMT) is

T = Lg - dz @ dL/d(dz) ; @ denotes the tensor product.

T = Lg + n dz@dz ; g = - u@u + dz@dz ; u = - dt as a 1-form;

T = n(u@u - dz@dz) + n dz@dz = \nu@udiv(T) = ==> na = ; where a = du(u) = acceleration =

as it must be, and div(\nu) = 0; we get back the eq. of continuity.

Next, introduce pressure in 2D spacetime. Look at the
transformation of coordinates from Cartesian (t,x) to
comoving (s,z). Now things become more interesting. The above
was mainly to establish notation and techniques. Now some physics.
Van Jacques

Van Jacques

Nov3-04, 09:57 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>Igor Khavkine wrote:\n> On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:\n>\n> >\n> >> On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:\n> >\n> >> > I recall Goldstein\'s "Classical Mechanics" does Lagrangian and\n> >> > Hamiltonian mechanics for particles, then tries but fails to do\n> >> > continua in the last chapter--a rather pathetic finish to an otherwise\n> >> > great text. One can now\n> >> > include continua and use differential geometry.\n> >\n> >> I\'m not sure how Goldstein "fails to do continua" in the last chapter.\n> >> I don\'t believe the treatment is erroneous, but it is brief compared to\n> >> the amount of space devoted to other topics.\n> >\n> > Goldstein fails in two respects:\n> > Since one wants to be able to introduce the EM field interacting with\n> > (continuous) matter in MHD and plasmas, one must treat everything\n> > relativistically (and have Lorentz invariance) from the start.\n> > Non-relativistic treatments create many problems, making the introduction\n> > of the EM field almost impossible, since the EM field is intrinsically\n> > relativistic. I wonder what physicists were thinking when trying to create\n> > Newtonian theories of MHD and plasmas. You can\'t mix one group of eqns.\n> > which are Lorentz invariant with another group that is Galilean invariant\n> > without creating a mess and introducing serious errors at the start.\n>\n> I don not think that doing magnetohydrodynamics was one of Goldstein\'s\n> goals for writing that chapter. Neither is his goal to do an exposition of\n> continuum mechanics. As best I can tell, the intent of the last chapter of\n> Goldstein\'s book is to introduce the reader to the generalities of the\n> formulation of classical field theory, as well as introduce some classical\n> fields that will later be used in relativistic quantum mechanics and QFT.\n\nThis is what he fails to do, IMO, and for the reasons I gave in\na post above. He mixes up the Lagrangian coordinates and the Cartesian\ncoordinates. The Lagrangian coords are the field, and the Cartesian\ncoords\nare the independent variables, as they are in EM and all other field\ntheories on spacetime. He tries to made the Lagragian coords the\nindependent\ncoordinates, and the Cartesian coords. into the fields, which leads\nhim to a mess and conceals important relations, and prevents the\ntheory from going anywhere.\n\n> Also, referring to your other post discussing Goldstein, I can only say\n> that his choice of notation has nothing to do with the validity of what he\n> is trying to say. And as far as I know, his treatment contains no serious\n\nIf you mean he manages to get the right non-rel eqns. in the end,\nits true that he does, but at great cost.\n\n> errors.\n> Also, his use of index and vector notation as opposed to\n> the more modern index-less one is no more a draw back than the use of\nthe\n> same notation for treating classical mechanics of point particles.\n\nIt can be done without vectors at all, which is how Maxwell\'s eqns.\nwere\nfirst written, in x,y,z components. Then with 3 vectors they had a\ngreat advance in notation, which made things compact,\nand helped them see relationships more easily.\nNow we have 4-vectors and differential geometry, which helps\nfurther. One can always reduce it to 3 + 1 or to 4 separate eqns.\n\nMisner, Thorne, and Wheeler\'s "Gravitation", Lightmann et al "A Problem\nBook In Relativity and Gravitation", the Choquet-Bruhat et al "Analysis\nManifold and Physics" are good examples to see how things can be better\nunderstood with these tools. Its not just notation. There are\nmany important new ideas in differntial geometry.\n\n> >> I\'ve found\n> >> that the use of differential geometry in physics is dictated more by the\n> >> culture of a given subfield instead of whatever is fashionable in modern\n> >> mathematics. Continuum mechanics can be done quite well with the more\n> >> orthodox use of vector analysis. In fact, this is what is done in most\n> >> texts on fluid dynamics and elasticity theory (for example the classic\n> >> texts by Landau and Lifshitz).\n\nBut these are old and need to be translated into new notation and new\nideas.\nSoper\'s "Classical Theory of Fields" contains many of the ideas that\nI am planning to present, and he uses 4-vectors, since it is all\nrelativistic, but he does it all with indices and no differential\ngeometry.\nIf you can find his book, it has a lot of excellent work which\nshould have been put in text, IMO.\nI have written up my work in papers which I would be happy to send\nto anyone who is interested.\n(which noone has yet read, and I hope to upload to the archives\nwhen I can find someone to vouch for me--I have a PhD and published\npapers in Ap J, but I don\'t think that is enough as I am no longer\nat a university).\n\n> > I thought that the tools of differential geometry and forms were well\n> > established by now. They included vector analysis, but they clarify\n> > everything. Esp. Stokes thm. and multiple integrals, which are especially\n> > important in dealing with continua.\n> >\n> > I gave the example of the EM field, which is a mess without the use of\n> > forms. Writing Maxwell\'s eqns. in 4D as dF = 0, F = dA, and d*F = *J, and\n> > p dimensional integrals over p-forms should be standard. I hope things\n> > like curl(A) and div(B) have been abandoned in favor of dA and *d*B. We\n> > can always project onto a 3D subspace of spacetime if necessary.\n>\n> I would disagree with your assumption. Basic differential geometry is not\n> part of the regular undergraduate curriculum for pretty much any\n> discipline, except perhaps mathematics. Vector analysis is, but it is hard\n> to make the connection between the two without prior exposure.\n\nI wouldn\'t expect them to necessarily be taught to undergrads. I was\nthinking of physics grad students and scientists.\n\n> I also take issue with your claim that since modern diff.geo. notation is\n> more compact and more elegant that it is more powerful. First of all,\n> "powerful" is not a well defined term. Instead one can look at its\n> utility. But before discussing utility, a specific purpose must be stated.\n> If the purpose is to express all the dynamical equations of your theory as\n> compactly and elegantly as possible? Then yes, I\'d say it\'s useful. If the\n> purpose is to perform a calculation in some geometry where space and time\n> are naturally split, then I\'d say that regular vector analysis is more\n> useful. If the purpose is to perform some numerical simulation, then\n> explicit coordinate choices must be made and everything must be calculated\n> in components. For this case I\'d say that the abstract index-less\n> formalism is one of the most useless.\n>\n> In conclusion, is modern diff.geo. notation universally useful? No. Is it\n> universally useless? Also no. Just as many other things, it falls\n> somewhere in between. To advocate the use of one notation or formalism\n> over another, the intended purpose must be stated and a strong argument\n> for the advantages for this particular purpose must be made.\n>\n> >> Although I still find your equations a little hard to decipher,\n> >\n> > What is hard to decipher? d = exterior derivative, /\\ = exterior product.\n> > I use \\x to denote the direct product in the EMT, but if you are familiar\n> > with classical field theory you should know how T is calculated from the\n> > Lagrangian. I will leave out \\x as I guess it doesn\'t help. I assume that\n> > one has been thru the field theory of the EM field. If not, this post\n> > won\'t make sense, but all physicists should be familiar with that.\n>\n> I find your equations hard to decipher because you make many claims\n> without justification that are not obviously true, although they very well\n> might be. At least they are not obvious to me. I\'ve taken a crack at\n> sorting some of your equations out in another branch of this thread, your\n> clarifications are welcome.\n\nI now understand what you mean, and will try to clarify.\nI have also sent you some email and a copy of the most important\npaper, which you should have by the time you read this.\n\n> >>I think all of this is well known by now, even if it is usually expressed\n> >>in different language.\n> >\n> > Is it? The only place I have seen it expressed in any language is in\n> > Soper\'s "Classical Field Theory", and he left a lot to be done.\n>\n> I don not know of where to find a thorough relativistic treatment of\n> continuum mechanics coupled to E&M. But some of this is worked out in\n> Chapter IV of Landau\'s Classical Field Theory (vol. 2) and Chapter XV of\n> his Hydrodynamics (vol. 6). Also, take a look at Chapter VIII of\n> Electrodynamics of Continuous Media by Landau and Lifshitz (vol. 8). There\n> the equations for non-relativistic MHD are worked out. You have a point\n> about possible pitfalls of using a Galilean theory for matter coupled to a\n> Lorentzian theory for the E&M field. However, the E&M equations can also\n> be reduced to a set of Galilean equations when the limit c -> oo is taken\n> into account. This amounts to dropping the terms corresponding to the\n> electric displacement current and the Faraday effect.\n\nI am aware of EM with c --> oo, but I don\'t like it. e.g.\n\nthe Alfven speed v^2_a = B^2/n non-rel vs B^2/(B^2 +n) relativistic.\nAs n --> 0, the 1st gives the incorrect v^2_a --> oo, the 2nd v^2_a-->\n1 = c.\nLichnerowitz, Andre has published a lot of excellent work on\nrelativistic\nmagnetohydrodynamics, and there are a few papers here and there\non continua with EM fields. But it is an area where there is still work\nto be done.\n\n> > The point I am making is that the theory is much more powerful in the\n> > field theoretic form. This is how things should be done in modern\n> > physics--it is how classical EM is done. The knowledge that A is the\n> > vector field for the field theory of EM, and that F = dA, and L = -\n> > (F|F)/2 is important and a powerful tool for problems in EM, as well as\n> > for integrating it with the rest of physics and for making progress.\n> >\n> > The same is true for continua. Everyone should know that the field theory\n> > of continua has 3 scalar fields, the Lagrangian coordinates, which come\n> > from the eqn. of continuity d*J = 0, and the Lagrangian L = - r + L_em,\n> >\n> > where r = total energy density, and L_em = Lagrangian for any EM fields.\n>\n> More powerful? That\'s an ambiguous statement that I discussed above.\n> I think it would be a safe guess that anyone working on a particular\n> theory knows what the relevant dynamical variables of that theory are.\n> But the choice of dynamical variables is not unique, you can also do a\n> change of variables. In E&M, all the equations can be expressed either in\n> terms of A or F. A is the variable with respect to which the variational\n> problem is constructed, but even then. I could define something like A\' =\n> *A, or A\' = A + dV, or A\' = (A|.), or other possibly non-linear field\n> redefinitions. Just as with E&M, all hydrodynamical equations can be\n> expressed in terms of the current density j or as you do in terms of the\n> Lagrangian coordinates, or perhaps Eulerian coordinates or some other\n> field definition. Once again, you must state your goal and argue why your\n> choice of field is best.\n>\n> Igor\n\nI agree, and hope my paper helps to convince you of some of what I say.\n\nI thank you again for taking an interest and spending the time\ngoing over this stuff. I hope you stick with it just a little\nwhile longer, when you have time anyway.\nIf nothing else it is good practice in relativistic thinking and\nmechanics. Most of this so far is trying to lay some groundwork.\nThis is the result of many years of work by me, and I found it all\nto be rewarding. I hope you will too.\n\nVan\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>Igor Khavkine wrote:
> On Wed, 27 Oct 2004 15:56:18 +0000, Van Jacques wrote:
>
> >
> >> On 2004-10-22, Van Jacques <vanja...@yahoo.com> wrote:
> >
> >> > I recall Goldstein's "Classical Mechanics" does Lagrangian and
> >> > Hamiltonian mechanics for particles, then tries but fails to do
> >> > continua in the last chapter--a rather pathetic finish to an otherwise
> >> > great text. One can now
> >> > include continua and use differential geometry.
> >
> >> I'm not sure how Goldstein "fails to do continua" in the last chapter.
> >> I don't believe the treatment is erroneous, but it is brief compared to
> >> the amount of space devoted to other topics.
> >
> > Goldstein fails in two respects:
> > Since one wants to be able to introduce the EM field interacting with
> > (continuous) matter in MHD and plasmas, one must treat everything
> > relativistically (and have Lorentz invariance) from the start.
> > Non-relativistic treatments create many problems, making the introduction
> > of the EM field almost impossible, since the EM field is intrinsically
> > relativistic. I wonder what physicists were thinking when trying to create
> > Newtonian theories of MHD and plasmas. You can't mix one group of eqns.
> > which are Lorentz invariant with another group that is Galilean invariant
> > without creating a mess and introducing serious errors at the start.
>
> I don not think that doing magnetohydrodynamics was one of Goldstein's
> goals for writing that chapter. Neither is his goal to do an exposition of
> continuum mechanics. As best I can tell, the intent of the last chapter of
> Goldstein's book is to introduce the reader to the generalities of the
> formulation of classical field theory, as well as introduce some classical
> fields that will later be used in relativistic quantum mechanics and QFT.

This is what he fails to do, IMO, and for the reasons I gave in
a post above. He mixes up the Lagrangian coordinates and the Cartesian
coordinates. The Lagrangian coords are the field, and the Cartesian
coords
are the independent variables, as they are in EM and all other field
theories on spacetime. He tries to made the Lagragian coords the
independent
coordinates, and the Cartesian coords. into the fields, which leads
him to a mess and conceals important relations, and prevents the
theory from going anywhere.

> Also, referring to your other post discussing Goldstein, I can only say
> that his choice of notation has nothing to do with the validity of what he
> is trying to say. And as far as I know, his treatment contains no serious

If you mean he manages to get the right non-rel eqns. in the end,
its true that he does, but at great cost.

> errors.
> Also, his use of index and vector notation as opposed to
> the more modern index-less one is no more a draw back than the use of
the
> same notation for treating classical mechanics of point particles.

It can be done without vectors at all, which is how Maxwell's eqns.
were
first written, in x,y,z components. Then with 3 vectors they had a
great advance in notation, which made things compact,
and helped them see relationships more easily.
Now we have 4-vectors and differential geometry, which helps
further. One can always reduce it to 3 + 1 or to 4 separate eqns.

Misner, Thorne, and Wheeler's "Gravitation", Lightmann et al "A Problem
Book In Relativity and Gravitation", the Choquet-Bruhat et al "Analysis
Manifold and Physics" are good examples to see how things can be better
understood with these tools. Its not just notation. There are
many important new ideas in differntial geometry.

> >> I've found
> >> that the use of differential geometry in physics is dictated more by the
> >> culture of a given subfield instead of whatever is fashionable in modern
> >> mathematics. Continuum mechanics can be done quite well with the more
> >> orthodox use of vector analysis. In fact, this is what is done in most
> >> texts on fluid dynamics and elasticity theory (for example the classic
> >> texts by Landau and Lifshitz).

But these are old and need to be translated into new notation and new
ideas.
Soper's "Classical Theory of Fields" contains many of the ideas that
I am planning to present, and he uses 4-vectors, since it is all
relativistic, but he does it all with indices and no differential
geometry.
If you can find his book, it has a lot of excellent work which
should have been put in text, IMO.
I have written up my work in papers which I would be happy to send
to anyone who is interested.
(which noone has yet read, and I hope to upload to the archives
when I can find someone to vouch for me--I have a PhD and published
papers in Ap J, but I don't think that is enough as I am no longer
at a university).

> > I thought that the tools of differential geometry and forms were well
> > established by now. They included vector analysis, but they clarify
> > everything. Esp. Stokes thm. and multiple integrals, which are especially
> > important in dealing with continua.
> >
> > I gave the example of the EM field, which is a mess without the use of
> > forms. Writing Maxwell's eqns. in 4D as dF = 0, F = dA, and d*F = *J, and
> > p dimensional integrals over p-forms should be standard. I hope things
> > like curl(A) and div(B) have been abandoned in favor of dA and *d*B. We
> > can always project onto a 3D subspace of spacetime if necessary.
>
> I would disagree with your assumption. Basic differential geometry is not
> part of the regular undergraduate curriculum for pretty much any
> discipline, except perhaps mathematics. Vector analysis is, but it is hard
> to make the connection between the two without prior exposure.

I wouldn't expect them to necessarily be taught to undergrads. I was
thinking of physics grad students and scientists.

> I also take issue with your claim that since modern diff.geo. notation is
> more compact and more elegant that it is more powerful. First of all,
> "powerful" is not a well defined term. Instead one can look at its
> utility. But before discussing utility, a specific purpose must be stated.
> If the purpose is to express all the dynamical equations of your theory as
> compactly and elegantly as possible? Then yes, I'd say it's useful. If the
> purpose is to perform a calculation in some geometry where space and time
> are naturally split, then I'd say that regular vector analysis is more
> useful. If the purpose is to perform some numerical simulation, then
> explicit coordinate choices must be made and everything must be calculated
> in components. For this case I'd say that the abstract index-less
> formalism is one of the most useless.
>
> In conclusion, is modern diff.geo. notation universally useful? No. Is it
> universally useless? Also no. Just as many other things, it falls
> somewhere in between. To advocate the use of one notation or formalism
> over another, the intended purpose must be stated and a strong argument
> for the advantages for this particular purpose must be made.
>
> >> Although I still find your equations a little hard to decipher,
> >
> > What is hard to decipher? d = exterior derivative, /\ = exterior product.
> > I use \x to denote the direct product in the EMT, but if you are familiar
> > with classical field theory you should know how T is calculated from the
> > Lagrangian. I will leave out \x as I guess it doesn't help. I assume that
> > one has been thru the field theory of the EM field. If not, this post
> > won't make sense, but all physicists should be familiar with that.
>
> I find your equations hard to decipher because you make many claims
> without justification that are not obviously true, although they very well
> might be. At least they are not obvious to me. I've taken a crack at
> sorting some of your equations out in another branch of this thread, your
> clarifications are welcome.

I now understand what you mean, and will try to clarify.
I have also sent you some email and a copy of the most important
paper, which you should have by the time you read this.

> >>I think all of this is well known by now, even if it is usually expressed
> >>in different language.
> >
> > Is it? The only place I have seen it expressed in any language is in
> > Soper's "Classical Field Theory", and he left a lot to be done.
>
> I don not know of where to find a thorough relativistic treatment of
> continuum mechanics coupled to E&M. But some of this is worked out in
> Chapter IV of Landau's Classical Field Theory (vol. 2) and Chapter XV of
> his Hydrodynamics (vol. 6). Also, take a look at Chapter VIII of
> Electrodynamics of Continuous Media by Landau and Lifshitz (vol. 8). There
> the equations for non-relativistic MHD are worked out. You have a point
> about possible pitfalls of using a Galilean theory for matter coupled to a
> Lorentzian theory for the E&M field. However, the E&M equations can also
> be reduced to a set of Galilean equations when the limit c -> oo is taken
> into account. This amounts to dropping the terms corresponding to the
> electric displacement current and the Faraday effect.

I am aware of EM with c --> oo, but I don't like it. e.g.

the Alfven speed v^{2_a} = B^2/n non-rel vs B^2/(B^2 +n) relativistic.
As n --> 0, the 1st gives the incorrect v^{2_a} --> oo, the 2nd v^{2_a}-->
1 = c.
Lichnerowitz, Andre has published a lot of excellent work on
relativistic
magnetohydrodynamics, and there are a few papers here and there
on continua with EM fields. But it is an area where there is still work
to be done.

> > The point I am making is that the theory is much more powerful in the
> > field theoretic form. This is how things should be done in modern
> > physics--it is how classical EM is done. The knowledge that A is the
> > vector field for the field theory of EM, and that F = dA, and L = -
> > (F|F)/2 is important and a powerful tool for problems in EM, as well as
> > for integrating it with the rest of physics and for making progress.
> >
> > The same is true for continua. Everyone should know that the field theory
> > of continua has 3 scalar fields, the Lagrangian coordinates, which come
> > from the eqn. of continuity d*J = 0, and the Lagrangian L = - r + L_{em},
> >
> > where r = total energy density, and L_{em} = Lagrangian for any EM fields.
>
> More powerful? That's an ambiguous statement that I discussed above.
> I think it would be a safe guess that anyone working on a particular
> theory knows what the relevant dynamical variables of that theory are.
> But the choice of dynamical variables is not unique, you can also do a
> change of variables. In E&M, all the equations can be expressed either in
> terms of A or F. A is the variable with respect to which the variational
> problem is constructed, but even then. I could define something like A' =
> *A, or A' = A + dV, or A' = (A|.), or other possibly non-linear field
> redefinitions. Just as with E&M, all hydrodynamical equations can be
> expressed in terms of the current density j or as you do in terms of the
> Lagrangian coordinates, or perhaps Eulerian coordinates or some other
> field definition. Once again, you must state your goal and argue why your
> choice of field is best.
>
> Igor

I agree, and hope my paper helps to convince you of some of what I say.

I thank you again for taking an interest and spending the time
going over this stuff. I hope you stick with it just a little
while longer, when you have time anyway.
If nothing else it is good practice in relativistic thinking and
mechanics. Most of this so far is trying to lay some groundwork.
This is the result of many years of work by me, and I found it all
to be rewarding. I hope you will too.

Van

Van Jacques

Nov3-04, 09:58 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>Arnold Neumaier wrote:\n\n> The secret is that most people like to answer questions that\n> fall into their field of expertise, if it does not take too\n\nI took a look at Goldstein\'s treatment of continua,\nand it is even worse than I recalled.\nHe is not the one who interchanges the Lagrangian and\nCartesian coords. That is other authors, like\nC. Eckart, R. Dewar, and others who have tried for\nsolns. to this problem. At least they get the eqns. of\nmotion. Goldstein can\'t even obtain the eqn. of motion for\na perfect fluid, na + grad(p) = 0. He gets eqns. for sound waves.\nHe should have left this out of his text, which\nis excellent in other areas, IMO.\n\nHere is the problem that most authors have\nwhen trying to do the field theory of fluids (or any continua):\nfor a single particle, the trajectory is x^i(t) ; i = 1,2,3\nt = time x^0 or proper time s.\nThis works for particles both rel and non-rel.\n\nBut for continua, we can\'t have x^i be fields, since they\nare now independent Cartesian coords.\nThe fields are z^i(x) ; i = 1,2,3; x = (x^a); a = 0,1,2,3\nThe Lagrangian L depends on all 3 z^i(x) and their partial derivs.\ndz/dx^a. L = L(z^i, dz^i; x)\n\nIt turns out that the fields are the 3 Lagrangian coordinates\n= the initial positions of the fluid particles,\nwhich are 3 scalar functions on spacetime. See Soper\nand my paper. (Jacques, on request).\n\nAny field theory on spacetime, whether relativistic or\nnon-relativistic, must be able to answer the following questions.\n\nWhat is the field(s) for the field theory?\nWhat is the Lagrangian, and how does it depend on the fields?\n\nI answer these in my paper, do perfect fluids, MHD, and plasma,\nand waves in these media. Thus I get the eqns. of motion\nincluding the EMT of the waves, and the eqns. governing\nthe waves in these media. Much of this can\'t be found\nanywhere else in the literature.\n\nVan Jacques\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>Arnold Neumaier wrote:

> The secret is that most people like to answer questions that
> fall into their field of expertise, if it does not take too

I took a look at Goldstein's treatment of continua,
and it is even worse than I recalled.
He is not the one who interchanges the Lagrangian and
Cartesian coords. That is other authors, like
C. Eckart, R. Dewar, and others who have tried for
solns. to this problem. At least they get the eqns. of
motion. Goldstein can't even obtain the eqn. of motion for
a perfect fluid, na + grad(p) = . He gets eqns. for sound waves.
He should have left this out of his text, which
is excellent in other areas, IMO.

Here is the problem that most authors have
when trying to do the field theory of fluids (or any continua):
for a single particle, the trajectory is x^i(t) ; i = 1,2,3
t = time x^0 or proper time s.
This works for particles both rel and non-rel.

But for continua, we can't have x^i be fields, since they
are now independent Cartesian coords.
The fields are z^i(x) ; i = 1,2,3; x = (x^a); a = 0,1,2,3
The Lagrangian L depends on all 3 z^i(x) and their partial derivs.
dz/dx^a. L = L(z^i, dz^i; x)

It turns out that the fields are the 3 Lagrangian coordinates
= the initial positions of the fluid particles,
which are 3 scalar functions on spacetime. See Soper
and my paper. (Jacques, on request).

Any field theory on spacetime, whether relativistic or
non-relativistic, must be able to answer the following questions.

What is the field(s) for the field theory?
What is the Lagrangian, and how does it depend on the fields?

I answer these in my paper, do perfect fluids, MHD, and plasma,
and waves in these media. Thus I get the eqns. of motion
including the EMT of the waves, and the eqns. governing
the waves in these media. Much of this can't be found
anywhere else in the literature.

Van Jacques

Doug Sweetser

Nov3-04, 10:02 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 Van Jacques:\n\n> We also know that the Lagrangian is\n>\n> L = - (F|F)/2 + (K|A) ;\n\nThis does not look correct to me. The current coupling term should have\nthe same sign as the field strength tensor contraction (Jackson, eq\n12.85). For your Lagrangian, like charges would attract. I once was\nmade to feel very foolish because I had made a similar sign error. Of\ncourse, I may also be in error because I am more familiar with indices\nthat a "|" which looks non-standard to me.\n\nDoing things relativistically is very appealing, but the end result of\nthe calculation applies only if the system involves particles moving\nrelativistically. For most systems, one needs to break spacetime\nsymmetry at some point to get to real-world data. The canonical\nexample was Schrodinger\'s relativistic wave equation which did not\nconnect to the data about the classical hydrogen atom until he made the\nwave equation non-relativistic with only one time derivative. You want\nto keep things relativistic for as long as possible, others bail out\nearlier.\n\nGupta and Bleuler quantized the EM field relativistically in 1950.\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 Van Jacques:

> We also know that the Lagrangian is
>
> L = - (F|F)/2 + (K|A) ;

This does not look correct to me. The current coupling term should have
the same sign as the field strength tensor contraction (Jackson, eq
12.85). For your Lagrangian, like charges would attract. I once was
made to feel very foolish because I had made a similar sign error. Of
course, I may also be in error because I am more familiar with indices
that a "|" which looks non-standard to me.

Doing things relativistically is very appealing, but the end result of
the calculation applies only if the system involves particles moving
relativistically. For most systems, one needs to break spacetime
symmetry at some point to get to real-world data. The canonical
example was Schrodinger's relativistic wave equation which did not
connect to the data about the classical hydrogen atom until he made the
wave equation non-relativistic with only one time derivative. You want
to keep things relativistic for as long as possible, others bail out
earlier.

Gupta and Bleuler quantized the EM field relativistically in 1950.
doug
quaternions.com

Igor Khavkine

Nov4-04, 03:37 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>"Van Jacques" <vanjac12@yahoo.com> wrote in message news:<1098911576.395769.122930@f14g2000cwb.googleg roups.com>...\n\n> Igor Khavkine wrote:\n> > I\'ve mentioned in a previous post that I have trouble interpreting\n> your\n> > notation and conclusions. Let me try to interpret what I understand\n> and\n> > perhaps you can fill in some blanks.\n> >\n> > On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:\n> >\n> > > Now the point of the post. What about continua. What is the\n> homogenous\n> > > eqn. which gives the field for continua? It is the eqn. of\n> continuity. Let\n> > > j = *J = n*u, where J = nu = matter current density, n = (rest)\n> mass\n> > > density, and u = 4-velocity.\n> >\n> > The first point is that I do not think it is a trivial fact that\n> > current density is represented by a 3-form in 4D spacetime. Second, I\n> am\n> > confused by your use of the Hodge star. You seem to be claiming that\n> J\n> > is a 1-form which is proportional to u, where u is a 4-velocity\n> vector.\n> > I guess the identification can be made with the metric, but this step\n> > cannot be omitted.\n>\n> I am glad you want that kind of rigor, but the metric does indeed\n> provide\n> and isomorphism between 4-vector fields and 1-forms. The way I was\n> taught was that this isomorphism is reflected in the fact that one\n> can use the metric to raise and lower indices. Since I didn\'t bother\n> with indices in this broad overall picture, I didn\'t think it\n> necessary to talk about the isomorphism between 1-forms and 4-vectors.\n>\n> Assuming there is no gravity so that space is flat, we have\n>\n> J^a = n u^a ; u^a = dx^a/ds ; s = proper time ; a = 0,1,2,3\n>\n> = *d*J = *dj\n>\n> J_a = g_ab J^b with g_ab = diag( -1,1,1,1)\n>\n> but to readers of sci.physics.research, whom I assume are physics grad\n>\n> students or PhDs, I don\'t think this kind of thing is necessary.\n> Continuing, the current density 3-form is j = *J, or\n\nThis is just what I suspected. I agree with you that the vector-1-form\nduality need not be mentioned every time it is invoked, but should be\nmentioned at least once for the sake of consistency if not clarity. BTW,\nI wouldn\'t be surprised if a large number of s.p.r readers never attended\ngrad school.\n\n> > For my own benefit and that of anyone interested, let me sketch that\n> a\n> > current density in an (k+1)-dimensional spacetime must indeed be an\n> > k-form. Start off in familiar territory and assume that space and\n> time\n> > are split.\n>\n> I don\'t like to split space and time until the end. The calculations\n> are easier in 4D, and many relationships are clear in 4D that become\n> invisible in 3 + 1 notation. e.g. dF = 0 ==> F = dA is clearer and\n> easier\n> in 4D, but a mess in 3 + 1 notation.\n\nIt\'s true that often 4D notation becomes simpler, however my point was\nthat it is not a priori obvious what are the right quantities and\nequations that are to be used in 4D. Fortunately, some simple physical\nreasoning and introduction of 3+1 coordinates allows us to derive\n4D equations that reduce to the expected result when a time direction\nis singled out. The power of tensors comes in here because once we\nknow what a tensor is in any given coordinate system, we know what\nit is in any coordinate system.\n\n> > Let n\' be the particle density (n\' integrated over a k-volume\n> > of space gives a number, must be a k-form on space).\n>\n> Again, lets use only Lorentz invariant quantities, like\n> the proper number density n in J = nu. Everything should be coordinate\n> independent, i.e. tensors (the term tensor includes Lorentz scalars.\n> When ever one has a current _density, it is really a 3-form, as you\n> say.\n> Note that we can then write n = sqrt[- (J|J)] ; which is obviously a\n> scalar because of the scalar product (J|J = g(J,J).\n>\n> > Let j\' be the particle\n> > current density. Given a (k-1)-dimensional hypersurface, the integral\n> of\n> > j\' over it also gives a number, the flux through the hypersurface per\n> > unit time. Hence j\' must be a (k-1)-form on space. The continuity\n> > equation takes the form (where V is a k-volume of space, and @V its\n> > boundary)\n> >\n> > @/@t int_V n = int_(@V) j\n>\n> Very good. Yes, in 3 + 1 form, as it is called.\n>\n> > or equilvalently in integral form\n> >\n> > int_(V at t2) n\' - int_(V at t1) n\' = int_(t1 to t2) int_(@V) j\' dt\n> > = int_(@Vx[t1,t2]) j\' /\\ dt .\n> >\n> > Now we have two k-forms on spacetime, n\' and j\' /\\ dt, let me denote\n> > their sum by j = n\' + j\' /\\ dt. Looking at the above equation\n> carefully,\n> > we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the\n> > boundary of the spacetime volume Vx[t1,t2]. Now the continuity\n> equation\n> > can be rewritten as\n> >\n> > int_@(Vx[t1,t2]) j = 0 which implies int_(Vx[t1,t2]) dj = 0.\n> >\n> > And since I can build any spacetime volume out of little pieces that\n> > look like Vx[t1,t2], the differential form of the continuity equation\n> > is just dj = 0.\n> >\n> > The above discussion justifies two things. First that the current\n> > density is given by a k-form j in a (k+1)-dimensional space time.\n> > Integrated over a space-like k-volume, j gives the number of\n> particles\n> > contained in it.\n>\n> Exactly. You have made me very happy by this post. Thank you.\n> You are someone I would be delighted to talk with about anything at\n> any time. If I can be of any help, feel free to ask me.\n> You can email me at vanjac12ATyahoo.com (replace AT with @).\n\nI take this as a complement of highest sort. :-)\n\n> I would say that if V is a 3D volume, or in fact any 3D hypersurface in\n> 4D, then\n>\n> N = Int_V(j) = total # of particles (or equivalently the rest mass)\n> in V at time x^0 = t,\n> or if it is a sphere moving thru time, then it is the flux of fluid\n> thru the sphere per unit time, as I think you said.\n>\n> > While integrated over a timelike k-hypersurface, j\n> > gives the total number of particles that have passed through it.\n> Second\n> > it justifies the simple dj = 0 form of the continuity equation in\n> > space-time. I might have messed up with the sign of the second term\n> in\n> > j, perhaps it is dt /\\ j, but I think the general idea is fine.\n>\n> Yes, you have it.\n>\n> > Now, about the relation between the velocity field u and the current\n> > density j. They can\'t be simply related by a Hodge transformation,\n> since\n> > one is a form and the other is a vector field. And how does the\n> > application of the Hodge star follow from first principles anyway?\n>\n> The 3-form dual to the 4-velocity (thought of as a 4-vector or a\n> 1-form,\n> it doesn\'t matter), is the 3D volume element\n> *u = d^3(x) = dx^1 /\\ dx^2 /\\ dx^3\n> orthogonal to the time direction defined by the 4-velocity u = d/ds\n> (recall that in a local comoving coord. system, u = @/@x^0 = @/@t,\n> using @ for the partial derivative (I usually write d for the partial\n> deriv. if it won\'t create confusion).\n\nAh I see it now. This is a nice way of thinking about the Hodge dual.\nWe expect the flux density through a surface to be (nu dot N) times\nthe surface volume element, with N being the surface unit normal vector.\nAnd this is exactly what j = *(nu) is.\n\n> > > Then write the eqn. of continuity div(J) = 0\n> > > as dj = 0. The exterior derivative of the 3-form j = 0, so that j\n> is\n> > > closed and therefore exact. Because of the isotropy of 3D space, we\n> have j\n> > > is the exterior product of the exterior derivative of 3 scalar\n> fields z^1,\n> > > z^2, z^3.\n> > >\n> > > j = dz^1 /\\ dz^2 /\\ dz^3 = *J\n>\n> Never mind that for now. Its a sloppy way of justifying something\n> that thakes some experience to see.\n\nPerhaps once more basic issues are settled you can give a hint why\nyou expect the above equation to be true in higher dimensions.\n\n> The thing you need to do now is to do the problem in explicit detail\n> in 1 space dimension, i.e. 2D spacetime. This allows one to get a grip\n> on everything.\n>\n> Let u^0 = dx^0/ds ; u^1 = dx^1/ds ; J = nu in 2D.\n>\n> Then the metric is g = diag(-1, 1), and the Hodge * operator is the\n> 2D Levi-Civita tensor with non-zero components\n>\n> e_01 = 1 = - e_10 ; e = dx^0 /\\ dx^1 is the 2D volume element\n>\n> (just as e is the 4D vol. element in 4D).\n>\n> j = *J = e(J) = e_ab J^a e^b ; where e^b = dx^b ; e_a = d/dx^a\n>\n> from differential geometry. Now,\n>\n> dj = 0 ==> j = dz = dz/dx^0 e^0 + dz/dx^1 e^1 = z_0 dx^0 + z_1 dx^1\n>\n> to get J^0 = ng = dz/dx^1 ; J^1 = ngv = - dz/dx^0\n>\n> Where u^0 = g = 1/sqrt(1 - v^2) ; u^1 = gv ; v = dx^1/dx^0 =\n> 3-velocity.\n>\n> I usually write (x^0, x^1) --> (t,x) ; so v = dx/dt, etc.\n>\n> Then for cold dust, L = - n = - sqrt(j|j) = - sqrt(dz|dz)\n> is the Lagrangian, and\n> classical field theory gives\n>\n> T = Lg - dz @ dL/d(dz) = - ng + dz@dz/n; @ denotes the tensor product.\n>\n> But j = *J = n*u = dz, so dz@dz/n = n*u@*u (again, these come after\n>\n> a lot of experience fooling around with this stuff).\n> Also, it turns out that\n>\n> g = - u @ u + *u @ *u ==> T = nuu ; div(T) = u[div(nu)] + n (u|grad)u\n>\n> = 0 + na = 0, ==> a = acceleration = 0 as we expect in this case.\n>\n> One can then move on to 4D, perfect fluids, MHD, plasma, waves in\n> all these media, the whole of the mechanics of continua can now\n> be formulated as a classical field theory, once we finally know what\n> the fields are.\n>\n> In 4D there 3 scalar field z^i instead of just the one, z(x).\n\nI am once again somewhat confused about what you are trying to\ncalculate here. First, it is once again not clear to me why the\nfluid lagrangian would take the form you suggest above. Second,\nwhy are you trying to compute the energy momentum tensor? What\ndoes it give us? The first thing I would try to do is calculate\nthe equations of motion and see how they are related to the equations\none would expect for a fluid. So I will try to address these issues\nbelow.\n\nFirst, consider the action for a single point particle\n\nS_a = int m_a ds_a.\n\nThe integral is over part of the worldline of the particle, ds_a is\nthe proper time element and m_a is its mass. If we have a collection\nof particles, then the action of this system is\n\nS = sum_a S_a = int (sum_a m_a ds_a).\n\nAt the moment I have many world lines each with its own proper time\ncoordinate s_a. As the number of particles goes to infinity and their\ndensity grows, the particle labels a translate in a sense to coordinate\nlabels and m_a translates to the particle mass density n. What we need\nto do is convert the above sum into an integral over spacetime as\nthe collection of particles tends to a continuum.\n\nTo do this, note from previous discussion that the integral of the\nflux density j over a spacelike surface gives the amount of mass\npresent in it if viewed as a surface of constant time. The sum\nsum_a m_a over the particles passing through the same surface will\ngive us the _proper_ mass contained in that surface. Hence the sum\nof m_a will approach the integral of j only if the spacelike surface\nin question is orthogonal to the velocity vector u_a for each particle.\nIn other words, the limit of sum_a m_a becomes int_S j, where S is\nspacelike surface everywhere orthogonal to u, the velocity vector\nfield of the continuum. We rewrite the action for the continuum as\n\nS = int (int_S j) ds,\n\nwhere the individual proper time coordinates s_a have been parametrized\nby the single proper time coordinate s comoving with the spacelike\nsurfaces S. The last thing to do is to turn the above expression into\na 4D one independent of any particular parametrization. For this note that\nthe 1-form ds representing the proper time element along the worldline\nof a particular point of the continuum is metric dual to the velocity\nfield u. In other words, treating u as a 1-form, we have\n\nS = int j /\\ u, (or maybe u /\\ j, depending on the orientation)\n\nwhere integration is now over any space-time volume. It is possible to\nrewrite the above in terms of j only. Recall the relation j = *(nu)\nor j = n *u, since n is a scalar field, and finally u = *j/n. The\naction acquires a very simple form, which even lacks square roots,\n\nS = int (j /\\ *j)/n.\n\nSince j /\\ *j = (j|j)vol, and (j|j) = n^2, my expression for the action\nreduces to yours. Note that I\'m cannot take 1/n out of the integral\nbecause I cannot say that n is a constant. In fact, it is a constant\nonly for the case of the so called incompressible fluid. In a more\ncomplicated case, we must supplement the above action principle with\nan equation of state specifying n based on some other properties of\nthe fluid.\n\nFrom this action principle we can derive the equations of motion.\nFrom the equation of continuity dj = 0, we can write j = dz, where\nz is a (k-1)-form if we are in a k+1D spacetime. As you point out\nin 1+1D, z is just a scalar field. The action must now be rewritten\nin terms of z\n\nS = 1/n int dz /\\ *dz,\n\nwhere I\'ve assumed n to be constant for simplicity (an incompressible\nfluid). The reason for using z is that it introduces derivatives into\nthe action, and that the variational equations derived for z will not\nbe trivial like they would be for j. Note the similarity to the\nelectromagnetic case, where the action is\n\nS_{EM} = int F /\\ *F = int dA /\\ *dA, (module constant prefactors).\n\nExcept that A is always a 1-form, while as I\'ve mentioned z must be a\n(k-1) form. In 2+1D, z will be a 1-form also so 2+1D E&M must be\nveri similar to 2+1D mechanics of an incompressible fluid. Interesting...\n\nRight, now it is easy to derive the equations of motion by minimizing\nS with respect to variations in z. The equations are simply\n\nd*dz = 0.\n\nLet me see if I calculate the explicit form of these equations in 1+1D.\nLet me choose coordinates t and x, in which the metric is diagonal (-1,1)\nand the volume form is dt/\\dx. The action of the Hodge dual is determined\nby dt/\\*dt = (dt|dt) dt/\\dx = -dt/\\dx, and dx/\\*dx = (dx|dx) dt/\\dx\n= dt/\\dx. In other words *dt = -dx, and *dx = -dt.\n\ndz = @z/@t dt + @z/@x dx,\n*dz = - @z/@t dx - @z/@x dt,\nd*dz = (- @^2 z/@t^2 + @^2 z/@x^2) dt/\\dx.\n\n=> @^2 z/@t^2 - @^2 z/@x^2 = 0.\n\nThe equation of motion is just the wave equation in 1+1D.\n\nAll of this is very interesting. And it seems you are posting new\nmaterial much faster than I can read it. I will try to go through\nthe rest when I have time. Also, to complete the picture of this\ntheory I\'d like to see how the equations of motion derived above\ncompare to the equations for relativistic fluid mechanics as give\nfor instance in Chapter XV of Landau\'s Hydrodynamics (vol 6). I\nhaven\'t addressed your calculation of the energy momentum tensor,\nbut I\'ll try in another post. For instance, I think your calculation\nis a bit old fashioned and perhaps ambiguous. The best way to calculate\nthe EMT seems to be to vary the action with respect to the metric.\n\nIgor\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>"Van Jacques" <vanjac12@yahoo.com> wrote in message news:<1098911576.395769.122930@f14g2000cwb.googlegroups. com>...

> Igor Khavkine wrote:
> > I've mentioned in a previous post that I have trouble interpreting
> your
> > notation and conclusions. Let me try to interpret what I understand
> and
> > perhaps you can fill in some blanks.
> >
> > On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:
> >
> > > Now the point of the post. What about continua. What is the
> homogenous
> > > eqn. which gives the field for continua? It is the eqn. of
> continuity. Let
> > > j = *J = n*u, where J = \nu = matter current density, n = (rest)
> mass
> > > density, and u = 4-velocity.
> >
> > The first point is that I do not think it is a trivial fact that
> > current density is represented by a 3-form in 4D spacetime. Second, I
> am
> > confused by your use of the Hodge star. You seem to be claiming that
> J
> > is a 1-form which is proportional to u, where u is a 4-velocity
> vector.
> > I guess the identification can be made with the metric, but this step
> > cannot be omitted.
>
> I am glad you want that kind of rigor, but the metric does indeed
> provide
> and isomorphism between 4-vector fields and 1-forms. The way I was
> taught was that this isomorphism is reflected in the fact that one
> can use the metric to raise and lower indices. Since I didn't bother
> with indices in this broad overall picture, I didn't think it
> necessary to talk about the isomorphism between 1-forms and 4-vectors.
>
> Assuming there is no gravity so that space is flat, we have
>
> J^a = n u^a ; u^a = dx^a/ds ; s = proper time ; a = 0,1,2,3
>
> = *d*J = *dj
>
> J_a = g_{ab} J^b with g_{ab} = diag( -1,1,1,1)
>
> but to readers of sci.physics.research, whom I assume are physics grad
>
> students or PhDs, I don't think this kind of thing is necessary.
> Continuing, the current density 3-form is j = *J, or

This is just what I suspected. I agree with you that the vector-1-form
duality need not be mentioned every time it is invoked, but should be
mentioned at least once for the sake of consistency if not clarity. BTW,
I wouldn't be surprised if a large number of s.p.r readers never attended
grad school.

> > For my own benefit and that of anyone interested, let me sketch that
> a
> > current density in an (k+1)-dimensional spacetime must indeed be an
> > k-form. Start off in familiar territory and assume that space and
> time
> > are split.
>
> I don't like to split space and time until the end. The calculations
> are easier in 4D, and many relationships are clear in 4D that become
> invisible in 3 + 1 notation. e.g. dF = ==> F = dA is clearer and
> easier
> in 4D, but a mess in 3 + 1 notation.

It's true that often 4D notation becomes simpler, however my point was
that it is not a priori obvious what are the right quantities and
equations that are to be used in 4D. Fortunately, some simple physical
reasoning and introduction of 3+1 coordinates allows us to derive
4D equations that reduce to the expected result when a time direction
is singled out. The power of tensors comes in here because once we
know what a tensor is in any given coordinate system, we know what
it is in any coordinate system.

> > Let n' be the particle density (n' integrated over a k-volume
> > of space gives a number, must be a k-form on space).
>
> Again, lets use only Lorentz invariant quantities, like
> the proper number density n in J = \nu. Everything should be coordinate
> independent, i.e. tensors (the term tensor includes Lorentz scalars.
> When ever one has a current _density, it is really a 3-form, as you
> say.
> Note that we can then write n = \sqrt[- (J|J)] ; which is obviously a
> scalar because of the scalar product (J|J = g(J,J).
>
> > Let j' be the particle
> > current density. Given a (k-1)-dimensional hypersurface, the integral
> of
> > j' over it also gives a number, the flux through the hypersurface per
> > unit time. Hence j' must be a (k-1)-form on space. The continuity
> > equation takes the form (where V is a k-volume of space, and @V its
> > boundary)
> >
> > @/@t \int_V n = \int_(@V) j
>
> Very good. Yes, in 3 + 1 form, as it is called.
>
> > or equilvalently in integral form
> >
> > \int_(V at t2) n' - \int_(V at t1) n' = \int_(t1 to t2) \int_(@V) j' dt
> > = \int_(@Vx[t1,t2]) j' /\ dt .
> >
> > Now we have two k-forms on spacetime, n' and j' /\ dt, let me denote
> > their sum by j = n' + j' /\ dt. Looking at the above equation
> carefully,
> > we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the
> > boundary of the spacetime volume Vx[t1,t2]. Now the continuity
> equation
> > can be rewritten as
> >
> > \int_@(Vx[t1,t2]) j = which implies \int_(Vx[t1,t2]) dj = .
> >
> > And since I can build any spacetime volume out of little pieces that
> > look like Vx[t1,t2], the differential form of the continuity equation
> > is just dj = .
> >
> > The above discussion justifies two things. First that the current
> > density is given by a k-form j in a (k+1)-dimensional space time.
> > Integrated over a space-like k-volume, j gives the number of
> particles
> > contained in it.
>
> Exactly. You have made me very happy by this post. Thank you.
> You are someone I would be delighted to talk with about anything at
> any time. If I can be of any help, feel free to ask me.
> You can email me at vanjac12ATyahoo.com (replace AT with @).

I take this as a complement of highest sort. :-)

> I would say that if V is a 3D volume, or in fact any 3D hypersurface in
> 4D, then
>
> N = \Int_V(j) = total # of particles (or equivalently the rest mass)
> in V at time x^0 = t,
> or if it is a sphere moving thru time, then it is the flux of fluid
> thru the sphere per unit time, as I think you said.
>
> > While integrated over a timelike k-hypersurface, j
> > gives the total number of particles that have passed through it.
> Second
> > it justifies the simple dj = form of the continuity equation in
> > space-time. I might have messed up with the sign of the second term
> in
> > j, perhaps it is dt /\ j, but I think the general idea is fine.
>
> Yes, you have it.
>
> > Now, about the relation between the velocity field u and the current
> > density j. They can't be simply related by a Hodge transformation,
> since
> > one is a form and the other is a vector field. And how does the
> > application of the Hodge star follow from first principles anyway?
>
> The 3-form dual to the 4-velocity (thought of as a 4-vector or a
> 1-form,
> it doesn't matter), is the 3D volume element
> *u = d^3(x) = dx^1 /\ dx^2 /\ dx^3
> orthogonal to the time direction defined by the 4-velocity u = d/ds
> (recall that in a local comoving coord. system, u = @/@x^0 = @/@t,
> using @ for the partial derivative (I usually write d for the partial
> deriv. if it won't create confusion).

Ah I see it now. This is a nice way of thinking about the Hodge dual.
We expect the flux density through a surface to be (\nu dot N) times
the surface volume element, with N being the surface unit normal vector.
And this is exactly what j = *(\nu) is.

> > > Then write the eqn. of continuity div(J) =
> > > as dj = . The exterior derivative of the 3-form j = 0, so that j
> is
> > > closed and therefore exact. Because of the isotropy of 3D space, we
> have j
> > > is the exterior product of the exterior derivative of 3 scalar
> fields z^1,
> > > z^2, z^3.
> > >
> > > j = dz^1 /\ dz^2 /\ dz^3 = *J
>
> Never mind that for now. Its a sloppy way of justifying something
> that thakes some experience to see.

Perhaps once more basic issues are settled you can give a hint why
you expect the above equation to be true in higher dimensions.

> The thing you need to do now is to do the problem in explicit detail
> in 1 space dimension, i.e. 2D spacetime. This allows one to get a grip
> on everything.
>
> Let u^0 = dx^0/ds ; u^1 = dx^1/ds ; J = \nu in 2D.
>
> Then the metric is g = diag(-1, 1), and the Hodge * operator is the
> 2D Levi-Civita tensor with non-zero components
>
> e_{01} = 1 = - e_{10} ; e = dx^0 /\ dx^1 is the 2D volume element
>
> (just as e is the 4D vol. element in 4D).
>
> j = *J = e(J) = e_{ab} J^a e^b ; where e^b = dx^b ; e_a = d/dx^a
>
> from differential geometry. Now,
>
> dj = ==> j = dz = dz/dx^0 e^0 + dz/dx^1 e^1 = z_0 dx^0 + z_1 dx^1
>
> to get J^0 = ng = dz/dx^1 ; J^1 = ngv = - dz/dx^0
>
> Where u^0 = g = 1/\sqrt(1 - v^2) ; u^1 = gv ; v = dx^1/dx^0 =
> 3-velocity.
>
> I usually write (x^0, x^1) --> (t,x) ; so v = dx/dt, etc.
>
> Then for cold dust, L = - n = - \sqrt(j|j) = - \sqrt(dz|dz)
> is the Lagrangian, and
> classical field theory gives
>
> T = Lg - dz @ dL/d(dz) = - ng + dz@dz/n; @ denotes the tensor product.
>
> But j = *J = n*u = dz, so dz@dz/n = n*u@*u (again, these come after
>
> a lot of experience fooling around with this stuff).
> Also, it turns out that
>
> g = - u @ u + *u @ *u ==> T = nuu ; div(T) = u[div(\nu)] + n (u|grad)u
>
> = + na = 0, ==> a = acceleration = as we expect in this case.
>
> One can then move on to 4D, perfect fluids, MHD, plasma, waves in
> all these media, the whole of the mechanics of continua can now
> be formulated as a classical field theory, once we finally know what
> the fields are.
>
> In 4D there 3 scalar field z^i instead of just the one, z(x).

I am once again somewhat confused about what you are trying to
calculate here. First, it is once again not clear to me why the
fluid lagrangian would take the form you suggest above. Second,
why are you trying to compute the energy momentum tensor? What
does it give us? The first thing I would try to do is calculate
the equations of motion and see how they are related to the equations
one would expect for a fluid. So I will try to address these issues
below.

First, consider the action for a single point particle

S_a = \int m_a ds_a[/itex].

The integral is over part of the worldline of the particle, ds_a is
the proper time element and m_a is its mass. If we have a collection
of particles, then the action of this system is

S = sum_a S_a = \int (sum_a m_a ds_a).

At the moment I have many world lines each with its own proper time
coordinate s_a. As the number of particles goes to infinity and their
density grows, the particle labels a translate in a sense to coordinate
labels and m_a translates to the particle mass density n. What we need
to do is convert the above sum into an integral over spacetime as
the collection of particles tends to a continuum.

To do this, note from previous discussion that the integral of the
flux density j over a spacelike surface gives the amount of mass
present in it if viewed as a surface of constant time. The sum
sum_a m_a over the particles passing through the same surface will
give us the _proper_ mass contained in that surface. Hence the sum
of m_a will approach the integral of j only if the spacelike surface
in question is orthogonal to the velocity vector u_a for each particle.
In other words, the limit of sum_a m_a becomes \int_S j, where S is
spacelike surface everywhere orthogonal to u, the velocity vector
field of the continuum. We rewrite the action for the continuum as

S = \int (\int_S j) ds,

where the individual proper time coordinates s_a have been parametrized
by the single proper time coordinate s comoving with the spacelike
surfaces S. The last thing to do is to turn the above expression into
a 4D one independent of any particular parametrization. For this note that
the 1-form ds representing the proper time element along the worldline
of a particular point of the continuum is metric dual to the velocity
field u. In other words, treating u as a 1-form, we have

S = \int j /\ u, (or maybe u /\ j, depending on the orientation)

where integration is now over any space-time volume. It is possible to
rewrite the above in terms of j only. Recall the relation j = *(\nu)
or j = n *u, since n is a scalar field, and finally u = *j/n. The
action acquires a very simple form, which even lacks square roots,

S = \int (j /\ *j)/n.

Since j /\ *j = (j|j)vol, and (j|j) = n^2, my expression for the action
reduces to yours. Note that I'm cannot take 1/n out of the integral
because I cannot say that n is a constant. In fact, it is a constant
only for the case of the so called incompressible fluid. In a more
complicated case, we must supplement the above action principle with
an equation of state specifying n based on some other properties of
the fluid.

From this action principle we can derive the equations of motion.
From the equation of continuity dj = 0, we can write j = dz, where
z is a (k-1)-form if we are in a k+1D spacetime. As you point out
in 1+1D, z is just a scalar field. The action must now be rewritten
in terms of z

[itex]S = 1/n \int dz /\ *dz,

where I've assumed n to be constant for simplicity (an incompressible
fluid). The reason for using z is that it introduces derivatives into
the action, and that the variational equations derived for z will not
be trivial like they would be for j. Note the similarity to the
electromagnetic case, where the action is

S_{EM} = \int F /\ *F = \int dA /\ *dA, (module constant prefactors).

Except that A is always a 1-form, while as I've mentioned z must be a
(k-1) form. In 2+1D, z will be a 1-form also so 2+1D E&M must be
veri similar to 2+1D mechanics of an incompressible fluid. Interesting...

Right, now it is easy to derive the equations of motion by minimizing
S with respect to variations in z. The equations are simply

d*dz = .

Let me see if I calculate the explicit form of these equations in 1+1D.
Let me choose coordinates t and x, in which the metric is diagonal (-1,1)
and the volume form is dt/\dx. The action of the Hodge dual is determined
by dt/\*dt = (dt|dt) dt/\dx = -dt/\dx, and dx/\*dx = (dx|dx) dt/\dx= dt/\dx. In other words *dt = -dx, and *dx = -dt.dz = @z/@t dt + @z/@x dx,
*dz = - @z/@t dx - @z/@x dt,
d*dz = (- @^2 z/@t^2 + @^2 z/@x^2) dt/\dx.

=> @^2 z/@t^2 - @^2 z/@x^2 = .

The equation of motion is just the wave equation in 1+1D.

All of this is very interesting. And it seems you are posting new
material much faster than I can read it. I will try to go through
the rest when I have time. Also, to complete the picture of this
theory I'd like to see how the equations of motion derived above
compare to the equations for relativistic fluid mechanics as give
for instance in Chapter XV of Landau's Hydrodynamics (vol 6). I
haven't addressed your calculation of the energy momentum tensor,
but I'll try in another post. For instance, I think your calculation
is a bit old fashioned and perhaps ambiguous. The best way to calculate
the EMT seems to be to vary the action with respect to the metric.

Igor

Van Jacques

Nov5-04, 06:10 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>Doug Sweetser wrote:\n> Hello Van Jacques:\n> Hello Van Jacques:\n>\n> > We also know that the Lagrangian is\n>\n> > L = - (F|F)/2 + (K|A) ;\n>\n> This does not look correct to me. The current coupling term should have\n> the same sign as the field strength tensor contraction (Jackson, eq\n> 12.85). For your Lagrangian, like charges would attract. I once was\n> made to feel very foolish because I had made a similar sign error.\n\nI don\'t think anyone should feel foolish over a sign error, though\none has to get them right at some time, of course.\n> Of\n> course, I may also be in error because I am more familiar with indices\n> that a "|" which looks non-standard to me.\n\nI don\'t like to use indices unless I have to. It makes things\nlook like such a mess. The notation\n<a,b> usually denoted the "pairing" of a vector and its dual,\nlike the \'bra\' \'ket\' notation of QM; <A|B>.\n\nThe notation (a|b) = g(a,b) = g_ij a^i b^j is pretty common.\nThe Lagrangian;\nI looked it up, and found both + and - for the (K|A) term.\nIts a matter of convention really, since one could let\nF --> - F (as some do) or (A --> - A), in which case we still have\n\n(F|F) with a - sign, but the (K|A) changes sign.\n\nI don\'t agree that it makes like charge attract.\n\nHave you looked in some texts? I think you will find both signs\nfor this term. I did.\n\n> Doing things relativistically is very appealing, but the end result of\n> the calculation applies only if the system involves particles moving\n> relativistically.\n\nThis is wrong. Relativistic formulas apply at all speeds. It is\nthe Galilean or Newtonian formulas that fail for high speeds,\nbut my point is that there are other problems. It is simply\nthe wrong formalism, especially for classical mechanics, which\nunfolds in space and time--i.e. in spacetime, as Einstein taught us.\n\nHave you read my posts? I gave the example of the Alfven\nspeed from the MHD eqs. v_a^2 = B^2/n non-rel and\nv_a^2 = B^2/(nf + B^2) relativistically, where nf = r + p\nin my notation. What happens as the density n --> 0 ?\nThe non-rel. eq. blows up, while the rel. eq give the correct\nresult that v_a^2 --> 1 = speed of light.\n\nIs low density also relativistic? Well, only the rel. formula\ndeals with low density correctly. There are many other such\nexamples. I say again, it is absurd to mix EM fields and matter\nusing non-rel. eqs. The EM field is fundamentally relativistic,\nand if you want to consider matter interacting with EM, as with\nMHD or plasma, you can\'t use some Galilean form of the EM eqns.\n\nBut even without the EM field there are situations that come up\nthat are not high velocity, that can only be understood\nrelativistically.\nWave propagation gives an example, which I plan to post.\n\nI can\'t tell you how horrified I was when I learned MHD and\nplasmas as non-relativistic theories. I determined then\nto do them relativistically, and all my studies have born out\nthe correctness of that 1st impression.\n\n> For most systems, one needs to break spacetime\n> symmetry at some point to get to real-world data.\n\nI wouldn\'t say one has to "break spacetime symmetry", but I\nknow what you mean. All you have to do is to form the spatial\nprojection operator (which always depends on the velocity of\nthe observer--there is no universal separate time and space,\nwe know that from Einstein). If the observer\'s 3-velocity is v,\nhis 4-velocity is u = u^a e_a in a Cartesian basis e_a, a = 0,1,2,3;\nu^0 = g = 1/sqrt(1 - v^2), u^i = g v^i.\n\nThen form the observer\'s spatial projection operator\n\nP = g + u @ u (@ == tensor product), the metric g = diag(-1,1,1,1)\n\nin a Cartesian (or Minkowski) CS.\n\nThen P(u) = 0. We can write any vector as A = - (A|u) u + P(A)\n\nin terms of its components along and ortho to u, and the same\nfor tensors, as I do above.\n\nThen we can use P to get the spatial part of vector, and u\nto get the time part. I will be doing this is future posts,\nand I would be glad to send a copy of my paper which does\nthis, if you email me for it.\n\n> The canonical\n> example was Schrodinger\'s relativistic wave equation which did not\n> connect to the data about the classical hydrogen atom until he made\nthe\n> wave equation non-relativistic with only one time derivative. You\nwant\n> to keep things relativistic for as long as possible, others bail out\n> earlier.\n\nWhat about the Dirac eqn., and relativistic corrections to the\nH atom. This is a drawback of his QM, not a virtue or an essential\nfeature. I am not considering QM now. I want to do classical\ncontinua correctly first.\n\n> Gupta and Bleuler quantized the EM field relativistically in 1950.\n> doug\n> quaternions.com\nWhat about Feynam\'s work?\n\nVan\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>Doug Sweetser wrote:
> Hello Van Jacques:
> Hello Van Jacques:
>
> > We also know that the Lagrangian is
>
> > L = - (F|F)/2 + (K|A) ;
>
> This does not look correct to me. The current coupling term should have
> the same sign as the field strength tensor contraction (Jackson, eq
> 12.85). For your Lagrangian, like charges would attract. I once was
> made to feel very foolish because I had made a similar sign error.

I don't think anyone should feel foolish over a sign error, though
one has to get them right at some time, of course.
> Of
> course, I may also be in error because I am more familiar with indices
> that a "|" which looks non-standard to me.

I don't like to use indices unless I have to. It makes things
look like such a mess. The notation
<a,b> usually denoted the "pairing" of a vector and its dual,
like the 'bra' 'ket' notation of QM; <A|B>.

The notation (a|b) = g(a,b) = g_{ij} a^i b^j is pretty common.
The Lagrangian;
I looked it up, and found both + and - for the (K|A) term.
Its a matter of convention really, since one could let
F --> - F (as some do) or (A --> - A), in which case we still have

(F|F)[/itex] with a - sign, but the (K|A) changes sign.

I don't agree that it makes like charge attract.

Have you looked in some texts? I think you will find both signs
for this term. I did.

> Doing things relativistically is very appealing, but the end result of
> the calculation applies only if the system involves particles moving
> relativistically.

This is wrong. Relativistic formulas apply at all speeds. It is
the Galilean or Newtonian formulas that fail for high speeds,
but my point is that there are other problems. It is simply
the wrong formalism, especially for classical mechanics, which
unfolds in space and time--i.e. in spacetime, as Einstein taught us.

Have you read my posts? I gave the example of the Alfven
speed from the MHD eqs. v_a^2 = B^2/n non-rel and
v_a^2 = B^2/(nf + B^2) relativistically, where nf = r + p
in my notation. What happens as the density n --> ?
The non-rel. eq. blows up, while the rel. eq give the correct
result that v_a^2 --> 1 = speed of light.

Is low density also relativistic? Well, only the rel. formula
deals with low density correctly. There are many other such
examples. I say again, it is absurd to mix EM fields and matter
using non-rel. eqs. The EM field is fundamentally relativistic,
and if you want to consider matter interacting with EM, as with
MHD or plasma, you can't use some Galilean form of the EM eqns.

But even without the EM field there are situations that come up
that are not high velocity, that can only be understood
relativistically.
Wave propagation gives an example, which I plan to post.

I can't tell you how horrified I was when I learned MHD and
plasmas as non-relativistic theories. I determined then
to do them relativistically, and all my studies have born out
the correctness of that 1st impression.

> For most systems, one needs to break spacetime
> symmetry at some point to get to real-world data.

I wouldn't say one has to "break spacetime symmetry", but I
know what you mean. All you have to do is to form the spatial
projection operator (which always depends on the velocity of
the observer--there is no universal separate time and space,
we know that from Einstein). If the observer's 3-velocity is v,
his 4-velocity is u = u^a e_a in a Cartesian basis e_a, a = 0,1,2,3;u^0 = g = 1/\sqrt(1 - v^2), u^i = g v^i.

Then form the observer's spatial projection operator

P = g + u @ u (@ == tensor product), the metric g [itex]= diag(-1,1,1,1)

in a Cartesian (or Minkowski) CS.

Then P(u) = . We can write any vector as A = - (A|u) u + P(A)

in terms of its components along and ortho to u, and the same
for tensors, as I do above.

Then we can use P to get the spatial part of vector, and u
to get the time part. I will be doing this is future posts,
and I would be glad to send a copy of my paper which does
this, if you email me for it.

> The canonical
> example was Schrodinger's relativistic wave equation which did not
> connect to the data about the classical hydrogen atom until he made
the
> wave equation non-relativistic with only one time derivative. You
want
> to keep things relativistic for as long as possible, others bail out
> earlier.

What about the Dirac eqn., and relativistic corrections to the
H atom. This is a drawback of his QM, not a virtue or an essential
feature. I am not considering QM now. I want to do classical
continua correctly first.

> Gupta and Bleuler quantized the EM field relativistically in 1950.
> doug
> quaternions.com
What about Feynam's work?

Van

Doug Sweetser

Nov6-04, 11:07 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 Van Jacques:\n\n> (F|F) with a - sign, but the (K|A) changes sign.\n>\n> I don\'t agree that it makes like charge attract.\n\nThe Lagrange density is a scalar, and the scalar for the charge coupling\nterm must have the same sign as the antisymetric field strength tensor\ncontraction.\n\n\n> I can\'t tell you how horrified I was when I learned MHD and\n> plasmas as non-relativistic theories. I determined then\n> to do them relativistically, and all my studies have born out\n> the correctness of that 1st impression.\n\nI have not read about MHD, so I have little useful input there.\n\n\n> What about the Dirac eqn., and relativistic corrections to the\n> H atom. This is a drawback of his QM, not a virtue or an essential\n> feature. I am not considering QM now. I want to do classical\n> continua correctly first.\n\nSchrodinger found the spacetime symmetric equation first, but it did not\nagree with experimental data.\n>\n>> Gupta and Bleuler quantized the EM field relativistically in 1950.\n>> doug\n>> quaternions.com\n> What about Feynam\'s work?\n\nYou should know about the Gupta-bleuler quantization method since it is\ncompletely 4D. This approach hightlights a technical problem.\nQuantizing a 4D wave equation with only a spin 1 field leads to a\nscalar and longitudinal mode that are not physical.\n\nI\'m not sure how Feynman fits in. The Feynman diagrams are a graphical\ntool to keep track of perturbation series expansions of solutions to\nquantum field equations. Perturbation theory should apply to classical\nquantum mechancs and relativistic quantum, although the latter will be\ntrickier.\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 Van Jacques:

> (F|F) with a - sign, but the (K|A) changes sign.
>
> I don't agree that it makes like charge attract.

The Lagrange density is a scalar, and the scalar for the charge coupling
term must have the same sign as the antisymetric field strength tensor
contraction.

> I can't tell you how horrified I was when I learned MHD and
> plasmas as non-relativistic theories. I determined then
> to do them relativistically, and all my studies have born out
> the correctness of that 1st impression.

I have not read about MHD, so I have little useful input there.

> What about the Dirac eqn., and relativistic corrections to the
> H atom. This is a drawback of his QM, not a virtue or an essential
> feature. I am not considering QM now. I want to do classical
> continua correctly first.

Schrodinger found the spacetime symmetric equation first, but it did not
agree with experimental data.
>
>> Gupta and Bleuler quantized the EM field relativistically in 1950.
>> doug
>> quaternions.com
> What about Feynam's work?

You should know about the Gupta-bleuler quantization method since it is
completely 4D. This approach hightlights a technical problem.
Quantizing a 4D wave equation with only a spin 1 field leads to a
scalar and longitudinal mode that are not physical.

I'm not sure how Feynman fits in. The Feynman diagrams are a graphical
tool to keep track of perturbation series expansions of solutions to
quantum field equations. Perturbation theory should apply to classical
quantum mechancs and relativistic quantum, although the latter will be
trickier.

doug
quaternions.com