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Tim,
Thank you for posting that little paraphrase (Brun's
Theorem via Whittaker) because it's the kind of thing I don't tend to
seek out and in this case it's very interesting. It's interesting to me because
having the Rosennean perspective but very limited formal education into
"Analytical Dynamics of Particles and Rigid Bodies" (etc)... there are several
aspects to this whole situation which leap out at me. The main ones have to do
with the exactness of the numbers of "integrals" supposedly needed for a
complete determination of the three coordinates and three velocity
components of the three-body-problem. "there are no more
than the currently-known 10 algebraic
integrals (the "classical integrals" - six of motion of center of gravity, three of angular momentum, and one of energy). This falls short of the 18 integrals needed" [snip...] I question such exact numbers, in general, and I suggest that the missing pieces of the puzzle (which would be hard to quantify, to say the least) are the relational pieces. Another aspect of this which intrigues me is that Whittaker remarks
on "excess degrees of freedom"-- which means, I take it, that he can't account
for as many as there are, using his calculations. This situation is rife
with ironies, but this isn't really the place or the time, I suppose... In any
case, once again, my intuition is that the extra degrees of freedom have to do
with relational issues of context which have been dispensed with in the models
he's using for his calculations.
The third issue here is also about those "excess degrees of
freedom" that the system exhibits but which the calculations are mute about.
This goes back to the discussion we had on non-holonomic constraints and how a
complex system as a "maximally non-holonomically constrained system" has
infinitely more degrees of freedom than a totally unconstrained system or even a
minimally constrained system. The peculiar thing about non-holonomic constraints
(meaning "context-dependent" constraints) is that such constraints end up
changing themselves, in a roundabout way. This is what concrete numbers can't
represent and they likewise cannot approach the flexibility that the
constant interactions (of system with context with system; ad infinitum)
create.
So, the solution to the three-body-problem will be a relational
one. Are you going to give it a shot?
Judith
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