At 10:17 PM 2/18/2011, Rich Murray wrote:
does classical mechanics always fail to predict or retrodict for 3 or
more Newtonian gravity bodies? Rich Murray 2011.02.18
I think there is a misconception here. There isn't any true two-body
or three-body problem because there are far, far more than two or
three bodies in the universe!
We simplify problems by neglecting what is remote. So we might,
indeed, look at 3-body problems; some solutions are known that are
special cases, if I'm correct. As the attempt to predict extends into
the future, however, the results become more and more inaccurate,
except in stable special cases.
I don't recall description of the overall problem mentioned when I
was young, before chaos theory became well-known. The problem is
infinite sensitivity to initial conditions. In setting up an attempt
to predict behavior of a system, even when the laws of motion are
well-defined, it's necessary to specify the initial conditions, i.e.,
the position and velocity of the elements. Now, from the Uncertainty
Principle, we can only know these to a certain combined accuracy, the
product of the uncertainties cannot be less than a fixed value.
But surely that's only a tiny detail!
However, turns out, some physical systems are infinitely sensitive to
initial conditions. Real physical systems, some fairly simple ones.
Using math, start with one particular exact initial condition, and
you get one result. Start from something infinitesimally different,
you can get a radically different result.
In practice, this means that the future of a system cannot, in
general, be exactly predicted, and for long periods of time,
relatively, the inaccuracy can become gross. There is a lovely
youtube video showing a pendulum suspended over four magnets. If you
start from a particular starting position, hovering over which magnet
will the pendulum end up settling? Outside regions close to the
magnets, it turns out to be *unpredictable.* That's because one
cannot set the initial conditions *exactly* the same. You can't
predict the outcome even by a history of tries, by releasing the
pendulum again from the supposedly same spot. You can't make the spot
'same' enough. (Probably. There might exist some regions where the
outcome is predictable, besides the obvious ones over the attractors.)
http://www.youtube.com/watch?v=Qe5Enm96MFQ&feature=related
