```Newsgroups: alt.sci.planetary,sci.physics
From: henry@zoo.toronto.edu (Henry Spencer)
Subject: Re: Solution to Classical Three Body Problem - Sundman (1912) ????
Date: Mon, 4 Dec 1995 18:24:52 GMT
Organization: U of Toronto Zoology
Lines: 35

In article <49osec\$7k0@agate.berkeley.edu> ted@physics.berkeley.edu writes:
>> The answer must be that the 3-body solution does not converge uniformly in
>>some way, not merely that convergence is slow. A tiny perturbation may
>>produce dramatic differences much later on. Chaos, no?
>
>This may be true; I don't know.  I don't think that anything like this
>was part of the original theorem, though...

As I understand it, Sundman's solution really does converge properly,
just much too slowly.  There are two uses for a solution:  insight, which
requires a solution that is simple enough to understand and manipulate,
and calculation, which requires either finite equations or reasonably
rapid convergence.  Sundman's solution is a minor footnote in orbital
mechanics because it's too complex to provide insight and too slowly
converging for calculation.

Precisely which functions count as "simple" for purposes of solvability
proofs depends on who's doing the proving.  (For example, if I recall
correctly, the proof that there is no solution for the general order-N
polynomial where N>4 becomes invalid if elliptic functions are allowed.)
But not all infinite series correspond to anything that is recognizably
a simple function, and Sundman's are among the messy ones.

That being said, three-body systems *do* have a tendency to be unstable,
although not necessarily chaotic.  Numerical exploration indicates that
the vast majority of all three-body systems eject one body (usually the
smallest) and settle down to be two-body systems.  The major exception
is when two of the bodies are in close orbits around each other and the
third is in a much larger orbit around the pair; almost all apparently-
stable multi-body systems are "hierarchical" systems of this sort.  There
are some minor exceptions like the Trojan points which involve quite
specific initial conditions.
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