Physics of Chaos in Hamiltonian Systems

  • G. M. Zaslavsky
World Scientific: 1998. 268pp £30, $44

Remarkably, the basic models of motion in physics — ranging from Newton's law of gravitation to modern string theory — can all be formulated as Hamiltonian dynamical systems. Here there is a striking pairing between position coordinates (configuration variables) and their rates of change (momentum variables). Each such pair of variables constitutes a ‘degree of freedom’ of the system. Hamiltonian dynamics normally conserves energy, but this need not be the case; for example, a frictionless pendulum with an oscillating support is a nonconservative Hamiltonian system. Time dependence is considered a ‘half’ degree of freedom. The goal of Physics of Chaos is to explore the structures and motion of systems with one and a half degrees of freedom.

Hamiltonian dynamics can show extraordinary complexity when the number of degrees of freedom exceeds one. This complexity gives rise to ‘fractal’ images that are just as visually engaging as the famed Mandelbrot set, and its study is vital for understanding everything from chemical reaction rates to the stability of the Solar System.

Computer studies show that the motion consists of islands of stability in a sea of chaos. These islands can even mimic quasi-crystalline patterns. Some of our understanding of such systems can be traced back to Poincaré, but many questions remain. The major one addressed here is: can statistical techniques be applied to such chaotic motion? While these methods apply to ‘uniformly’ chaotic systems, the ubiquitous islands of stability lead to long correlations that contravene the use of standard statistical (for example, diffusive) techniques. The theory of transport in such systems relies on the geometry of the stable manifolds and of the ‘cantori’ (structures that act as nearly impermeable barriers) that permeate the chaotic sea; this has been carefully treated by Stephen Wiggins in Chaotic Transport in Dynamical Systems (Springer, 1991).

George Zaslavsky develops ‘fractional kinetics’ in an attempt to give a smoothed, but nondiffusive, description. This phenomenological description captures some aspects of the stickiness of islands, but I believe its mathematical justification remains elusive. Perhaps that is an excellent reason to read this book.