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offered a prize of 2,500 crowns for the answer to the question of whether or not the
solar system is stable. One of the respondents (and the eventual winner of this contest)
was Henri Poincaré (1854-1912). The King wanted to know whether the planets would
remain in their present orbits for eternity, or whether the Moon would escape from the
Earth or crash into it, or whether the Earth would meander into the Sun. This was and
still is one of the open questions in astronomy, not so much for its practical importance,
but because we ought to be able to answer the question. It is a testament to his genius for
analysis that Poincaré was able to win the prize without directly answering the question.
In 1890 Poincaré submitted a monograph establishing a new branch of mathematics,
or more properly a new way of analyzing an established branch of mathematics. He
used topological ideas to determine the general properties of dynamical equations. This
new tool was applied to the problem in celestial mechanics of many bodies moving
under mutual Newtonian gravitational attraction. He was able to show that the two-body
problem had periodic solutions in general, but that if a third body is introduced into this
universe no simple analytic solution could be obtained. The three-body problem could
not be solved by the methods of celestial mechanics! He was able to show that if the third
body had a lighter mass than the other two its orbit would have a very complex structure
and could
not
be described by simple analytic functions. One hundred years later it
was determined that this complicated orbit is fractal, but to Poincaré the complexity of
the three-body problem indicated the unanswerability of the King's question regarding
the stability of the solar system. For an excellent discussion see either Stewart [
41
]or
Ekeland [
19
], or, better yet, read both. Thus, even for the epitome of clockwork motion
given by the motion of the planets, unpredictability intrudes.
Somewhat later Poincaré [
38
] was able to articulate the general perspective resulting
from the failure to solve the three-body problem:
A very small cause which escapes our notice determines a considerable effect that we cannot fail
to see, and then we say that the effect is due to chance. If we knew exactly the laws of nature and
situation of the universe at the initial moment, we could predict exactly the situation of that same
universe at a succeeding moment. But even if it were the case that the natural laws had no longer
any secret for us, we could still only know the initial situation approximately. If that enables us
to predict the succeeding situation with the same approximation, that is all we require, and we
should say that the phenomenon had been predicted, that it is governed by laws. But it is not
always so: it may happen that small differences in the initial conditions produce very great ones
in the final phenomena. A small error in the former will produce an enormous error in the latter.
Prediction becomes impossible, and we have the fortuitous phenomenon.
Poincaré sees an intrinsic inability to make long-range predictions due to a sen-
sitive dependence of the evolution of a network on its initial state, even though the
network is deterministic. This was a departure from the view of Laplace, who believed
in strict determinism, and to his mind this implied absolute predictability. Uncertainty
for Laplace was a consequence of imprecise knowledge, so that probability theory was
necessitated by incomplete and imperfect observations.
It is Poincaré's “modern” view of mechanics, a view that took over 100 years for
the scientific community to adopt, if only in part, that we make use of in this chapter.
Moreover, we develop this view keeping in mind that our end goal extends far beyond
