AI and cognitive psychology rant (getting more and more OT - tell me if I should shut up)

Andrew Dalke adalke at mindspring.com
Sun Nov 2 18:32:24 CET 2003


Stephen Horne:
> Are you sure [about chaos in the solar system]?

> ... I thought the orbit of the planets around the sun
> had been proven stable. Which implies that you needn't worry about
> chaos unless you are worried about the minor deviations from the
> idealised orbits - the idealised bit can be treated as constant, and
> forms a very close approximation of reality no matter what timescale
> you are working in.

Quoting from Ian Stewart's "The Problems of Mathematics" (c) 1987,
which is a very good book, and I encourage people to get a copy:

  ..could the other planets resonate with the Earth .. to make it collide
  with Mars, or run off into the cold and empty interstellar wastes?
  For imaginary solar systems simpler than ours, this kind of thing
  can happen.  Could it happen to us?  In 1887 King Oscar II of
  Sweden offered a prixe of 2,500 crowns for an answer.
  ..
  As far as we know today, no closed form solutions exist [for the
  general N body problem with N > 2]; at any rate, the general
  behaviour is enormously complicated.... Lagrange and Laplace
  between them did manage to show that the total departure from
  circularity of the orbits of the planets in the Solar System is
  constant; and that this total, even if concentrated on the Earth,
  will not switch it to an orbit that escapes the Sun altogether, like
  that of a comet.  But this didn't show that the Earth might not
  slowly gain energy from other planets, and drift out by bigger
  and bigger almost-circles until we become lost in the silence and
  the dark.  Proof was still lacking that we will neither leave the
  hearth nor fall into the fire, and for that the king offered his crowns.
  [I do like Stewart's writing style :) ]

  [So Poincare' invented topology and the idea of phase space.]
  He did not settle the question of the stability of the solar system:
  that had to wait until the 1960s.  But he made such a dent in it
  that in 1889 he was awarded his coveted Oscar, and he throughly
  deserved it. .. For instance, he proved that in the motion of three
  bodies there are always infinitely many distinct periodic motions...
  ...
  In 1963, using extensive topological arguments, Kolmogorov,
  Vladimir Arnol'd and Jurgen Moser were able to respond to
  the question 'Is the Solar System stable?' with a resounding and
  definitive answer: 'Probably'.  Their method (usually called KAM
  Theory) shows that most of the possible motions are built up from
  a superposition of periodic ones.  The planets never exactly
  repeat their positions, but keep on almost doing so.  However,
  if you pick an initial set-up at random, there is a small but
  non-zero chance of following a different type of motion, whereby
  the system may lose a planet (or worse)--though not terribly
  quickly. ... The fasciniating point is that there is no way to tell
  by observation which of these two types of behaviour will occur.
  Take any configuration that leads to almost periodic motion; then
  there are configurations as close as you like where planets wander
  off.  Conversely, adjacent to any initial configuration from which
  a planet wanders off, there are others for which the motion is
  almost periodic.  The two types of behaviour are mixed up
  together like spaghetti and bolognese sauce.

                    Andrew
                    dalke at dalkescientific.com






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