Re: Field theory for continua

From: Igor Khavkine (k_igor_k_at_lycos.com)
Date: 10/28/04 

Date: Thu, 28 Oct 2004 18:28:40 +0000 (UTC)

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