Re: Field theory for continua

From: Van Jacques (vanjac12_at_yahoo.com)
Date: 11/03/04 

Date: Wed, 3 Nov 2004 15:58:18 +0000 (UTC)

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) = 0. 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

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