Chaos with Few Degrees of Freedom

Abstract

As a mathematician, astronomer, and physicist, Poincaré was the first to study the phenomenon of chaos in classical mechanics, in particular the three-body-problem of the Moon orbiting around the Earth under the perturbation of the Sun. His work helps us to understand atomic and molecular physics, because our intuition in these areas is always based on a classical picture for the motion of the nuclei and electrons. The Moon's motion was found to be multi-periodic by the early observers; Newton explained the main frequencies and calculated the lowest terms in the corresponding Fourier expansion. But inspite of ingenious efforts, like the work of Hill involving as a starting point a periodic orbit of the full problem, Poincaré showed that perturbation theory is unable to guarantee the convergence of the Fourier expansion. In this traditional representation, the lunar trajectory would be restricted to a four-dimensional invariant torus although the available phase-space has eight dimensions. The invariant tori are destroyed because of increasing phase-lock due to perturbations, whenever the system is close to a resonance. The KAM-theorem gives a mathematical criterion for this loss, which is best seen in a surface of section for such simple systems as the Anisotropic and the Diamagnetic Kepler Problem (AKP and DKP). Instead of invariant tori, the flow in phase space involves a double foliation where each trajectory is the intersection of one leaf from the unstable foliation with a leaf from the stable foliation. Neighboring trajectories drift exponentially away from each other in the future along the unstable leaf, and approach each another in the stable leaf coming from the past. This situation, called “hard chaos”, also allows for a symbolic description of each trajectory in terms of a simple code like binary sequences in the AKP. Einstein showed that the usual connection with quantum mechanics is valid only for the case of “regular behavior”, i.e. the presence of invariant tori. For classically chaotic systems, VanVleck's expression for the propagator seems to be the best starting point, provided it is summed over all the classical trajectories that go from the source to the defector. Its trace becomes a sum over periodic orbits, which lends additional weight to Poincaré's emphasis on their importance in classical mechanics. In trying to understand the trace of an operator as a function of time, or the scattering phase-shift as a function of momentum, one runs into almost-periodic functions. They are able to mimic locally any arbitrary smooth function provided the spectrum of frequencies, although discrete, is rich enough. This unexpected, but smooth behavior can be seen as a symptom of quantum chaos, in contrast to the fractal nature of classical chaos.