4
1 Overview
and hidden, and is integrable. A three body gravitational system is not integrable.
Newton’s derivation of Kepler’s laws was based on the properties of the two-body
system. However, the dynamical interactions of the many bodies that comprise the
solar system lead to deviations from the predictions of Kepler’s laws, and lead one
to ask why the solar system is, in fact, so regular. Is the solar system stable (Moser
1975)? Will it maintain its present configuration into the future? These questions
have not yet been fully answered.
Questions concerning the stability and the future evolution of the solar system
have occupied scientists and mathematicians for the past 300 years. Until computers
were invented, all mathematical theories used perturbation expansions of various
types. In the eighteenth century, important contributions were made by Euler,
Lagrange, and Laplace on predicting the change in the geometry of orbits due to
small perturbations and on determining the overall stability of orbits. In addition,
Lagrange (1889) reformulated Newtonian mechanics in terms of a variational
principle that vastly extended our ability to analyze the behavior of dynamical
systems and allowed a straight-forward extension to continuum mechanics.
In the nineteenth century, there were two very important pieces of work that
laid the groundwork for our current view of mechanics. Hamilton reformulated
mechanics (Hamilton 1940) so that the dynamics of a mechanical system could
be described in terms of a momentum-position phase space rather than a velocityposition phase space as is the case for the Lagrangian formulation. This step is
extremely important because in the Hamiltonian formulation (which describes the
evolution of mechanical systems in terms of coupled first-order differential equations) the flow of trajectories in phase space is volume-preserving. Furthermore, if
symmetries exist (such as the space-time symmetries), then some of the generalized
momenta of the system may be conserved, thus reducing the dimension of the phase
space in which we must work.
The relation between the symmetries of a system and conservation laws was first
clarified by Noether (1918). Noether’s work provides one of the most important
tools of twentieth-century science, because the key to much of what we are
able to predict in science is symmetry. Symmetries imply conservation laws, and
conservation laws give conservative classical mechanics and quantum mechanics
whatever predictive power they have. Conservation laws are even responsible for
the existence of thermodynamics and hydrodynamics.
Another extremely important piece of work in the nineteenth century was due
to Poincaré (1899). Poincaré not only closed the door on an era but created the
first crack in the facade of determinism. Before Poincaré, most work on dynamics,
subsequent to Newton, involved computation of deviations from KepIer-type orbits
for two massive bodies that are perturbed by a third body. The idea was to take a
Kepler orbit as a first approximation and then compute successive corrections to it
using perturbation theory. One must then show that the perturbation expansions thus
obtained converge.
The problem of whether or not perturbation series converge was so important that
it was the subject of a prize question posed by King Oscar II of Sweden in 1885. The
question read as follows: For an arbitrary system of mass points which attract each
1 Overview
and hidden, and is integrable. A three body gravitational system is not integrable.
Newton’s derivation of Kepler’s laws was based on the properties of the two-body
system. However, the dynamical interactions of the many bodies that comprise the
solar system lead to deviations from the predictions of Kepler’s laws, and lead one
to ask why the solar system is, in fact, so regular. Is the solar system stable (Moser
1975)? Will it maintain its present configuration into the future? These questions
have not yet been fully answered.
Questions concerning the stability and the future evolution of the solar system
have occupied scientists and mathematicians for the past 300 years. Until computers
were invented, all mathematical theories used perturbation expansions of various
types. In the eighteenth century, important contributions were made by Euler,
Lagrange, and Laplace on predicting the change in the geometry of orbits due to
small perturbations and on determining the overall stability of orbits. In addition,
Lagrange (1889) reformulated Newtonian mechanics in terms of a variational
principle that vastly extended our ability to analyze the behavior of dynamical
systems and allowed a straight-forward extension to continuum mechanics.
In the nineteenth century, there were two very important pieces of work that
laid the groundwork for our current view of mechanics. Hamilton reformulated
mechanics (Hamilton 1940) so that the dynamics of a mechanical system could
be described in terms of a momentum-position phase space rather than a velocityposition phase space as is the case for the Lagrangian formulation. This step is
extremely important because in the Hamiltonian formulation (which describes the
evolution of mechanical systems in terms of coupled first-order differential equations) the flow of trajectories in phase space is volume-preserving. Furthermore, if
symmetries exist (such as the space-time symmetries), then some of the generalized
momenta of the system may be conserved, thus reducing the dimension of the phase
space in which we must work.
The relation between the symmetries of a system and conservation laws was first
clarified by Noether (1918). Noether’s work provides one of the most important
tools of twentieth-century science, because the key to much of what we are
able to predict in science is symmetry. Symmetries imply conservation laws, and
conservation laws give conservative classical mechanics and quantum mechanics
whatever predictive power they have. Conservation laws are even responsible for
the existence of thermodynamics and hydrodynamics.
Another extremely important piece of work in the nineteenth century was due
to Poincaré (1899). Poincaré not only closed the door on an era but created the
first crack in the facade of determinism. Before Poincaré, most work on dynamics,
subsequent to Newton, involved computation of deviations from KepIer-type orbits
for two massive bodies that are perturbed by a third body. The idea was to take a
Kepler orbit as a first approximation and then compute successive corrections to it
using perturbation theory. One must then show that the perturbation expansions thus
obtained converge.
The problem of whether or not perturbation series converge was so important that
it was the subject of a prize question posed by King Oscar II of Sweden in 1885. The
question read as follows: For an arbitrary system of mass points which attract each