1.2 Historical Overview
5
other according to Newton’s laws, assuming that no two points ever collide, give the
coordinates of the individual points for all time as the sum of a uniformly convergent
series whose terms are made up of known functions (Moser 1975). Poincaré entered
the contest and won the prize by showing that such series could be expected to
diverge because of small denominators caused by internal resonances.
We now know that resonances, that give rise to these small divisors, are
associated with the onset of chaos. Because of these divergences, it appears to be
impossible to make long-time predictions concerning the evolution of mechanical
systems (with a few exceptions such as the two-body Kepler system) using
perturbation expansions.
No further progress was made on the problem of long-time prediction in
mechanics until l954 when Kolmogorov (1954) outlined a proof, for systems of the
type proposed in King Oscar’s question, that a majority of the trajectories are quasiperiodic and can be described in terms of a special type of perturbation expansion.
In 1962, Arnol’d (1963) constructed a formal proof of Kolmogorov’s results for
a three-body system with an analytic Hamiltonian, and Moser (1968) obtained a
similar result for twist maps. The result of the work of Kolmogorov, Arnol’d, and
Moser (KAM) is that series expansions describing the motion of some orbits in
many-body systems are convergent, provided the natural frequencies associated
with these orbits are not close to resonance. The work of Arnol’d, also showed
that nonintegrable systems, with three or more degrees of freedom, are intrinsically
unstable. They contain a dense web of resonance lines, the Arnol’d web, that allows
diffusion to occur throughout the available phase space. The question of how rapid
the diffusion will be depends on the parameters of the system.
Shenker and Kadanoff (1982) and MacKay (1983) were able to show that at the
parameter value at which a given KAM torus (with quadratic irrational winding
number) is destroyed, the rational approximates have self-similar structure and
the areas in phase space that they occupy are related by scaling laws. They also
showed that the rational approximates play a dominant role in the destruction of
KAM tori. Escande and Doveil (1981) developed a renormalization theory for the
destruction of KAM tori directly from the Hamiltonian for systems with two degrees
of freedom. Thus, Hamiltonian systems, much like equilibrium systems near a phase
transition, can exhibit self-similar structure.
Much of the behavior that occurs in classical systems also occurs in their
quantum counterpart. However, because of the Heisenberg uncertainty relations,
we are forced to describe classical and quantum systems from quite different
perspectives. In classical systems, we can examine the evolution of individual orbits
in phase space, and we can see directly the chaotic flow of trajectories in phase
space. If we were to describe the evolution of the classical system in terms of the
probability distribution in phase space, using the Liouville equation, we would have
to search for the signatures of chaos in the behavior of the probability distributions
and eigenvalues of the Liouville operator. This has been done for very simple chaotic
maps (Driebe 1999), but it is a formidable task when dealing with Newtonian
mechanical systems with two or more degrees of freedom.
5
other according to Newton’s laws, assuming that no two points ever collide, give the
coordinates of the individual points for all time as the sum of a uniformly convergent
series whose terms are made up of known functions (Moser 1975). Poincaré entered
the contest and won the prize by showing that such series could be expected to
diverge because of small denominators caused by internal resonances.
We now know that resonances, that give rise to these small divisors, are
associated with the onset of chaos. Because of these divergences, it appears to be
impossible to make long-time predictions concerning the evolution of mechanical
systems (with a few exceptions such as the two-body Kepler system) using
perturbation expansions.
No further progress was made on the problem of long-time prediction in
mechanics until l954 when Kolmogorov (1954) outlined a proof, for systems of the
type proposed in King Oscar’s question, that a majority of the trajectories are quasiperiodic and can be described in terms of a special type of perturbation expansion.
In 1962, Arnol’d (1963) constructed a formal proof of Kolmogorov’s results for
a three-body system with an analytic Hamiltonian, and Moser (1968) obtained a
similar result for twist maps. The result of the work of Kolmogorov, Arnol’d, and
Moser (KAM) is that series expansions describing the motion of some orbits in
many-body systems are convergent, provided the natural frequencies associated
with these orbits are not close to resonance. The work of Arnol’d, also showed
that nonintegrable systems, with three or more degrees of freedom, are intrinsically
unstable. They contain a dense web of resonance lines, the Arnol’d web, that allows
diffusion to occur throughout the available phase space. The question of how rapid
the diffusion will be depends on the parameters of the system.
Shenker and Kadanoff (1982) and MacKay (1983) were able to show that at the
parameter value at which a given KAM torus (with quadratic irrational winding
number) is destroyed, the rational approximates have self-similar structure and
the areas in phase space that they occupy are related by scaling laws. They also
showed that the rational approximates play a dominant role in the destruction of
KAM tori. Escande and Doveil (1981) developed a renormalization theory for the
destruction of KAM tori directly from the Hamiltonian for systems with two degrees
of freedom. Thus, Hamiltonian systems, much like equilibrium systems near a phase
transition, can exhibit self-similar structure.
Much of the behavior that occurs in classical systems also occurs in their
quantum counterpart. However, because of the Heisenberg uncertainty relations,
we are forced to describe classical and quantum systems from quite different
perspectives. In classical systems, we can examine the evolution of individual orbits
in phase space, and we can see directly the chaotic flow of trajectories in phase
space. If we were to describe the evolution of the classical system in terms of the
probability distribution in phase space, using the Liouville equation, we would have
to search for the signatures of chaos in the behavior of the probability distributions
and eigenvalues of the Liouville operator. This has been done for very simple chaotic
maps (Driebe 1999), but it is a formidable task when dealing with Newtonian
mechanical systems with two or more degrees of freedom.