The first field theory-continua [Archive]

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Van Jacques

Sep17-04, 05:32 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>\nThe theoretical basis of the mechanics of continua is the\nfirst field theory, and is often neglected. We can learn a lot\nabout fields from this theory.\n\nThe Lagrangian for cold matter is the 4-form\n\nL = - n dV, where n = proper number density, and dV = the 4D volume\n\nelement. The fields for continua are the Lagrangian coordinates\nz^i(x) = x^i(t = 0) which label the worldlines of the fluid elements.\n\n>From j = dz^1 /\\ dz^2 /\\ dz^3 = *J = *(nu), (where J = nu = matter\ncurrent),\n\nn^2 = (*J|*J) = (j|j). Consider 2D spacetime (t,x); then we have a\nsingle scalar field z(t,x), and j = dz, n^2 = - (dz/dt)^2 +(dz/dx)^2,\n\nThe equation of continuity is dj = 0 ==> j = dz, which implies\nthe existence of the scalar field z (as dF = 0 ==> F = dA in EM).\n\nThe existence and importance of the scalar fields z^i (there are 3\nwhen the motion is in 3D space) was neglected in my education\n(during the 60s and early 70s). I don\'t know how things are taught\nnow.\n\nThe energy momentum tensor is found as usual in (classical) field\ntheory;\nT = Lg - dz^i dL/d(dz^i) ; where L is a scalar (not a 4-form).\n\nThen div(T) = 0 gives the usual eqns. of motion.\n\nIts instructive to do this for an ideal fluid and MHD if you haven\'t\ndone it before.\n\nI am also writing up some work I did on this which I plan to make\navailable. I do the basic theory for fluids, MHD, and plasma, then\ndo a treatment of waves which includes the interaction of the\nwaves with the average flow.\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 theoretical basis of the mechanics of continua is the
first field theory, and is often neglected. We can learn a lot
about fields from this theory.

The Lagrangian for cold matter is the 4-form

L = - n[/itex] dV, where n = proper number density, and dV = the 4D volume

element. The fields for continua are the Lagrangian coordinates
z^i(x) = x^i(t = 0) which label the worldlines of the fluid elements.

>From j = dz^1 /\ dz^2 /\ dz^3 = *J = *(\nu), (where J = \nu = matter
current),

n^2 = (*J|*J) = (j|j). Consider 2D spacetime (t,x); then we have a
single scalar field z(t,x), and j = dz, [itex]n^2 = - (dz/dt)^2 +(dz/dx)^2,

The equation of continuity is dj = ==> j = dz, which implies
the existence of the scalar field z (as dF = ==> F = dA in EM).

The existence and importance of the scalar fields z^i (there are 3
when the motion is in 3D space) was neglected in my education
(during the 60s and early 70s). I don't know how things are taught
now.

The energy momentum tensor is found as usual in (classical) field
theory;
T = Lg - dz^i dL/d(dz^i) ; where L is a scalar (not a 4-form).

Then div(T) = gives the usual eqns. of motion.

Its instructive to do this for an ideal fluid and MHD if you haven't
done it before.

I am also writing up some work I did on this which I plan to make
available. I do the basic theory for fluids, MHD, and plasma, then
do a treatment of waves which includes the interaction of the
waves with the average flow.

Van Jacques

Sep19-04, 06:55 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>\nIn the field theory of continua, I argue that dj = 0 ==>\nj = dk, where j = *J = *(nu) is a 3-form dual to the matter\ncurrent J = nu, k a 2-form. (* = Hodge operator).\n\nI would like to make an argument analogous to that made in\nelectromagnetism (EM), that dF = 0 ==> F = dA, A = 1-form potential.\n\nThe argument must go something like this; dj = 0 ==> j = dk\n= a 3D spatial volume element, since j(u) = 0, i.e., j lies\nin the space orthogonal to the 4-velocity u.\n\nIn 2D spacetime, j = *J is a 1-form, and the eq. of\ncontinuity is dj = 0 ==> j = dz, where z a function.\n\nHow does one argue that in 4D j = dz^1 /\\ dz^2 /\\ dz^3 = dk ?\n\nI could write k = \\eps_{abc} z^a /\\ dz^b /\\ dz^c, but this\ndoesn\'t seem useful.\nI could say that the spatial part of spacetime is isotropic, so that if\n\nj = dz in 1 space dimension (j is the volume element in 1D),\nthen in 3D j = 3D volume element.\n\nOne could also argue that \\int_V{j} = M, where M = total rest mass\nor total # of particles, so that j must be the 3D volume element\nweighted by number density or rest mass density r, i.e.,\n\nj = r dz^1 /\\ dz^2 /\\ dz^3 = *J = *(nu), where n = sqrt(j|j)\n\n= proper number density. The r is not important. It\ncan be absorbed into the z^i, and r appears simply so\nthat the z^i can be a Cartesian coordinate system.\n\nThe eqn. of continuity is div(J) = *dj = 0, so it is analogous to\nMaxwell\'s homogeneous eq.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>In the field theory of continua, I argue that dj = ==>
j = dk, where j = *J = *(\nu) is a 3-form dual to the matter
current J = \nu, k a 2-form. (* = Hodge operator).

I would like to make an argument analogous to that made in
electromagnetism (EM), that dF = ==> F = dA, A = 1-form potential.

The argument must go something like this; dj = ==> j = dk= a 3D spatial volume element, since j(u) = 0, i.e., j lies
in the space orthogonal to the 4-velocity u.

In 2D spacetime, j = *J is a 1-form, and the eq. of
continuity is dj = ==> j = dz, where z a function.

How does one argue that in 4D j = dz^1 /\ dz^2 /\ dz^3 = dk ?

I could write k = \eps_{abc} z^a /\ dz^b /\ dz^c, but this
doesn't seem useful.
I could say that the spatial part of spacetime is isotropic, so that if

j = dz[/itex] in 1 space dimension (j is the volume element in 1D),
then in 3D j = 3D volume element.

One could also argue that \int_V{j} = M, where M = total rest mass
or total # of particles, so that j must be the 3D volume element
weighted by number density or rest mass density r, i.e.,

j = r dz^1 /\ dz^2 /\ dz^3 = *J = *(\nu), where [itex]n = \sqrt(j|j)

= proper number density. The r is not important. It
can be absorbed into the z^i, and r appears simply so
that the z^i can be a Cartesian coordinate system.

The eqn. of continuity is div(J) = *dj = 0, so it is analogous to
Maxwell's homogeneous eq.

Van

Sep29-04, 08:30 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>This shows how fluids, MHD, plasma, and waves should be done, IMO.\nIs this approach well known?\n==========\nConsider the simplest case for fluid motion; a cold fluid in 1\ndimension.\n\nThe Lagrangian L = - n = - (energy density of rest mass).\n\nThe scalar field for the canonical field theory is the Lagrangian\ncoordinate z(t,x) = initial position of the fluid element at x at time\nt:\n\nz(o,x) = x. Z can be thought of as a label for the fluid elements, i.e.\neach element is labeled by it initial position.\n\nThe total mass M = Int(r dz) = Int (n dx), where r(z) is the density in\nz, and\nn = proper density in a local comoving coord. system (CMCS).\n\nJ = nu = current 4-vector, u = 4-vel (here we have only 2 dimensions).\nLet j = *J where * denotes the Hodge dual.\n\nThe eq. of continuity div(J) = *(dj) = 0 ==> dj = 0 ==> j = dz\ngives us our scalar field (we use j = r dz so that z can be a Cartesian\ncoordinate).\n\nj = r dz = *J = *(nu) ;\nJ = *j = \\eps(j) = r(- z_0 e_1 + z_1e_0),\n\nz_a = dz/dx^a ; x^0 = t, x^1 = x, e^a = dx^a\n\nJ^0 = nu^0 = r z_1 ; J^1 = - r z_0\n\n- (J|J) = (j|j) = - n^2 (u|u) = n^2 = r^2(dz|dz) = r^2[- (z_0)^2 +\n(z_1)^2]\n\nso L = - n = - r sqrt(dz|dz). From L we get the canonical\nenergy-momentum\ntensor and the conservation equations:\n\nT = Lg - dz dL/ddz ; T_a^b = L \\delta_a^b - z_a dL/dz_b,\n\ng = diag(-1,1) = metric. dz dL/ddz = - n(g + uu), ==>\n\nT = nuu, --> div(T) = na + div(nu) u = na = 0\n\nwhere a = du(u) = relativistic acceleration = 0.\n\nIf spacetime were curved (gravity) we would still have a = 0,\nmotion along geodesics.\n\nThis can be generalized to 4D spacetime, magnetohydrodynamics, plasmas,\nother fluids, and waves.\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>This shows how fluids, MHD, plasma, and waves should be done, IMO.
Is this approach well known?
==========
Consider the simplest case for fluid motion; a cold fluid in 1
dimension.

The Lagrangian L = - n = - (energy density of rest mass).

The scalar field for the canonical field theory is the Lagrangian
coordinate z(t,x) = initial position of the fluid element at x at time
t:

z(o,x) = x. Z can be thought of as a label for the fluid elements, i.e.
each element is labeled by it initial position.

The total mass M = \Int(r dz) = \Int (n dx), where r(z) is the density in
z, and
n = proper density in a local comoving coord. system (CMCS).

J = \nu =[/itex] current 4-vector, u = 4-vel (here we have only 2 dimensions).
Let j = *J where * denotes the Hodge dual.

The eq. of continuity div(J) = *(dj) = ==> dj = ==> j = dz
gives us our scalar field (we use j = r dz so that z can be a Cartesian
coordinate).

j = r dz = *J = *(\nu) ;J = *j = \eps(j) = r(- z_0 e_1 + z_{1e_0}),z_a = dz/dx^a ; x^0 = t, x^1 = x, e^a = dx^aJ^0 = \nu^0 = r z_1 ; J^1 = - r z_0- (J|J) = (j|j) = - n^2 (u|u) = n^2 = r^2(dz|dz) = r^2[- (z_0)^2 +(z_1)^2]

so L = - n = - r \sqrt(dz|dz). From L we get the canonical
energy-momentum
tensor and the conservation equations:

T = Lg - dz dL/ddz ; T_a^b = L \delta_a^b - z_a dL/dz_b,g = diag(-1,1) = metric. dz dL/ddz = - n(g + uu), ==>

T = nuu, --> [itex]div(T) = na + div(\nu) u = na =

where a = du(u) = relativistic acceleration = .

If spacetime were curved (gravity) we would still have a = 0,
motion along geodesics.

This can be generalized to 4D spacetime, magnetohydrodynamics, plasmas,
other fluids, and waves.