Determinacy in classical physics (naive question)

From: Laurent Fortel (no.spam.please_at_free.fr)
Date: 07/17/04 

Date: Sat, 17 Jul 2004 19:33:00 +0200

I am no physicist, but a maths teacher trying to use his vacations time
to have a look again at the physics he learnt as an undergraduate...
Preliminary notice : I am definitely not reasoning as a physicist (a
theoretical physicist perhaps...) since physical reality is NOT the
subject of my question. Indeed my question is rather a maths question.

Try to Google with "classical physics" and "deterministic". Many sources
(most of them quantum mechanics elementary expositories) will assert
with no detail that "classical physics is deterministic". One at least
precises that it means the classical theory of gravitation +
electrodynamics
http://online.redwoods.cc.ca.us/DEPTS/science/chem/storage/Schrod/

What I am looking for is a precise (mathematically formalized, ideally)
proposition meaning that "classical physics is deterministic", and some
directions to look for a proof.

Here is a tentative compendium of what I understand presently, any help
to go further will be welcome.

1 - About gravitation :

1a) A finite set of massive particules interacting via gravitation, in a
newtonian space-time.

This is the only thing I fully understand. Elementary differential
equations theory asserts clearly that if I know the position and the
velocity of each particle at t=0, every position and velocity at t=t_0
are implied. No problem on this one.

1b) Gravitation in a continuous universe.

No idea whether this is a reasonable question (but it seemed to me an
exercice easier than struggling with Maxwell's equations...)

It seems easy to write down a non-relativistic gravitation theory
replacing the finite massive particules m_i by a distribution of volumic
mass mu(x,y,z,t), just adding down an equation for conservation of mass.
Doing it so, the data would be mu(x,y,z,0) and the velocity field
v(x,y,z,0) ; "deterministic" would mean that mu(x,y,z,t) and v(x,y,z,t)
are thence uniquely determined for each t. This reminds me of no maths I
  have met, but might be well-known. Any directions of reading ?

2 - About electrodynamics :

2b) Maxwell equations in a continuous universe (newtonian space-time
with aether, or special relativity that is the furthest I know...)

Here I think I can write down a list of variables describing the
universe at time t=0 (mu distribution of volumic mass ; rho distribution
of charges ; velocity field). Then I can write a list of equations
(Maxwell equations, Newton law linking derivatives of the velocity field
and the formula for force, conservation of mass (conservation of charge
being implied by Maxwell equations)). "Determinacy of classical
mechanics" would seem to mean that when you give mu, rho and v for t=0,
their values for all t are implied. But the maths underneath seem at
least tricky ; how can it be that such an assumption is so universally
"well-known" while its proof is so seldom shown ? Probably I have a
completely false idea of what "determinacy of classical
electromagnetism" mean. Help !

2a) (For memory only) Classical electrodynamics for a finite set of
massive and charged particles.

If I was able to write a set of equations describing that, like at 1a),
I would probably understand what their "determinacy" means, using very
simply differential equations theory. But I suppose this is hopeless, is
it ?

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