1.2 Historical Overview
3
in the Principia Newton proposed his universal inverse square law of gravitation. He
then used it to derive Kepler’s empirical laws of planetary motion, to account for the
motion of the moon and the phenomenon of tides, to explain the precession of the
equinoxes, and to account for the behavior of falling bodies in Earth’s gravitational
field.
The success and power of Newton’s laws led to a great optimism about our
ability to predict the behavior of mechanical objects and, as a consequence, led
to the huge growth in science that we see today. In addition, it was accompanied
by a deterministic view of nature that is perhaps best exemplified in the writings of
Laplace. In his Philosophical Essay on Probabilities, Laplace states (Laplace 1951):
Given for one instant an intelligence which could comprehend all the forces by
which nature is animated and the respective situation of the beings who compose it—
an intelligence sufficiently vast to submit these data to analysis—it would embrace
in the same formula the movements of the greatest bodies of the universe and those
of the lightest atom. For it, nothing would be uncertain and the future, as the past,
would be present before its eyes.
This deterministic view of nature was completely natural, given the success of
Newtonian mechanics, and it persists up until the present day. Newton’s three laws
of motion led to a description of the motion of point masses in terms of a set of
coupled second-order differential equations. The theory of extended objects can be
derived from Newton’s laws by treating them as collections of point masses. If we
can specify the initial velocities and positions of the point particles, then Newton’s
equations for the point particles (obtained from the second law) should determine all
past and future motion. However, we now know that the assumption that Newton’s
equations can predict the future is a fallacy. Newton’s equations are, of course, the
correct starting point of mechanics, but in general, they only allow us to determine
the long-time behavior of integrable mechanical systems, few of which can be found
in nature. Newton’s laws, for most systems, describe inherently random behavior
and cannot determine the future evolution of any real system (except for very short
times) in more than a probabilistic sense.
The belief that Newtonian mechanics is a basis for determinism was formally
laid to rest by Sir Lighthill (1986) in a lecture to the Royal Society on the threehundredth anniversary of Newton’s Principia. In his lecture, Lighthill says . . . I
speak . . . once again on behalf of the broad global fraternity of practitioners
of mechanics. We are all deeply conscious today that the enthusiasm of our
forebears for the marvelous achievements of Newtonian mechanics led them to
make generalizations in this area of predictability which, indeed, we may have
generally tended to believe before 1960, but which we now recognize were false.
We collectively wish to apologize for having misled the general educated public
by spreading ideas about the determinism of systems satisfying Newton’s laws of
motion that, after 1960, were to be proved incorrect . . . .
In a sense, Newton (and Western science) were fortunate because the solar system
has amazingly regular behavior considering its complexity, and one can predict its
short-time behavior with fairly good accuracy. Part of the reason for this is the
fact that the two-body Kepler system is governed by symmetries, both space-time
3
in the Principia Newton proposed his universal inverse square law of gravitation. He
then used it to derive Kepler’s empirical laws of planetary motion, to account for the
motion of the moon and the phenomenon of tides, to explain the precession of the
equinoxes, and to account for the behavior of falling bodies in Earth’s gravitational
field.
The success and power of Newton’s laws led to a great optimism about our
ability to predict the behavior of mechanical objects and, as a consequence, led
to the huge growth in science that we see today. In addition, it was accompanied
by a deterministic view of nature that is perhaps best exemplified in the writings of
Laplace. In his Philosophical Essay on Probabilities, Laplace states (Laplace 1951):
Given for one instant an intelligence which could comprehend all the forces by
which nature is animated and the respective situation of the beings who compose it—
an intelligence sufficiently vast to submit these data to analysis—it would embrace
in the same formula the movements of the greatest bodies of the universe and those
of the lightest atom. For it, nothing would be uncertain and the future, as the past,
would be present before its eyes.
This deterministic view of nature was completely natural, given the success of
Newtonian mechanics, and it persists up until the present day. Newton’s three laws
of motion led to a description of the motion of point masses in terms of a set of
coupled second-order differential equations. The theory of extended objects can be
derived from Newton’s laws by treating them as collections of point masses. If we
can specify the initial velocities and positions of the point particles, then Newton’s
equations for the point particles (obtained from the second law) should determine all
past and future motion. However, we now know that the assumption that Newton’s
equations can predict the future is a fallacy. Newton’s equations are, of course, the
correct starting point of mechanics, but in general, they only allow us to determine
the long-time behavior of integrable mechanical systems, few of which can be found
in nature. Newton’s laws, for most systems, describe inherently random behavior
and cannot determine the future evolution of any real system (except for very short
times) in more than a probabilistic sense.
The belief that Newtonian mechanics is a basis for determinism was formally
laid to rest by Sir Lighthill (1986) in a lecture to the Royal Society on the threehundredth anniversary of Newton’s Principia. In his lecture, Lighthill says . . . I
speak . . . once again on behalf of the broad global fraternity of practitioners
of mechanics. We are all deeply conscious today that the enthusiasm of our
forebears for the marvelous achievements of Newtonian mechanics led them to
make generalizations in this area of predictability which, indeed, we may have
generally tended to believe before 1960, but which we now recognize were false.
We collectively wish to apologize for having misled the general educated public
by spreading ideas about the determinism of systems satisfying Newton’s laws of
motion that, after 1960, were to be proved incorrect . . . .
In a sense, Newton (and Western science) were fortunate because the solar system
has amazingly regular behavior considering its complexity, and one can predict its
short-time behavior with fairly good accuracy. Part of the reason for this is the
fact that the two-body Kepler system is governed by symmetries, both space-time