From: ddixon@nyx.cs.du.edu (David Dixon)
Subject: Re: Compelling Mysteries (II)
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Keywords: compelling, mysteries, partridges and pear trees
Organization: Nyx, Public Access Unix at U. of Denver Math/CS dept.
Date: Wed, 11 Nov 92 17:45:54 GMT

In article <1992Nov10.151421.11274@murdoch.acc.Virginia.EDU> crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass) writes:
>
> (...lots of stuff...)
>
>         c) the long-time stability of the solar system
>     that several of us discussed earlier.  Biology tends to tell us
>     that the solar system (or at least the earth's orbit) has been stable
>     for long lengths of time.  Calculations and perturbation theory tend
>     to tell us differently.  Of course, I don't believe the calculations,
>     but what do I know?  Opinions are very cheap, so spill them all over the
>     place.  Where is our fundamental misunderstanding?  Where are others?

Applying perturbation theory to many-body (and other nasty non-linear
problems) can be naughty.  Perturbation theory relies on the ability to
truncate a solution when many of the terms are "small".  The discovery of
chaos has shown this to be a bogus approach in a large class of problems,
because the small terms can have big effects.  End result: you can't write
down closed form solutions, even approximately, and expect them to have
any relation to what is physically observed.

Now, whether or not this has anything to do with the solar system stability
problem, I don't know.  But it does give me an opportunity to leap upon my
soapbox :-)  A good portion of the program in science for the past several
hundred years has been something like: "Here's a problem.  Let's write down
some differential equations and try to crank out a solution."  Which then
people dutifully attempted to do, and are still doing today.  Unfortunately
for our erstwhile theoreticians, it turned out that except in a few nice
(and pretty boring) cases, finding these solutions seemed to be very hard.
"But," they said, "Since we know the solutions exist, we'll keep trying.  In
the meantime, let's develop lots of machinery for finding approximate 
solutions, since these will tell us to an arbitrary desired accuracy what
the *real* answer is."  And thus it has gone since Newton, or thereabouts.

Sadly, such a program is screwed from the start.  In analogy with the
Goedel (no umlaut, dammit) Incompleteness Theorem, it can be proved that
while solutions to DE's may exist, they most often can't be found in
closed form, i.e., written on a finite amount of paper.  Worse yet, it can
be shown that in the space of all such problems, those with closed form
solutions are not dense, while those with no closed form solution do form
a dense set.  So there's a mystery for you: given the above, what should
physics be working on now?  Because when you get right down to it, there's
an awful lot of classical mechanics that is not understood.

>     I like a bit of wonder and mystery in the universe, it gets so boring
>     when a common attitude is that everything is fundamentally solved.
>     And actually, I'm just looking for things to add to *my* list. 
>
>                                   dale bass
>
>-- 
A man after my own heart :-)  How about this:  What is the link between
the micro and macro worlds?  The correspondence principle only seems to
work in certain cases, e.g., the Schroedinger equation cannot be chaotic.

					Dave