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Judith,
I don't recall if
Whittaker himself used the term "excess degrees of freedom", I just know I've
seen it in other discussions elsewhere of the 3-body system.
I agree that the
missing information is bound up with the organization, the relational qualities,
of the system. As I mentioned previously, from within the state-based formalism,
those qualities cannot be described, and so shows up symptomatically as missing
information. I have been working on how to describe this missing information,
but as yet having nothing concrete to present. In the end, the dynamical
system will almost certainly still be unsolvable in exact form, since the
additional information will have to be a "complementary model" in some
non-state-based formalism and so would not add information directly into the
dynamical equations.
Regards,
Tim
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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