Introduction

Abstract

Long as it may have taken to realize, we are now certain that there are two radically different types of motion in classical Hamiltonian mechanics: the regular motion of integrable systems and the chaotic motion of nonintegrable systems. The harmonic oscillator and the Kepler problem show regular motion, while systems as simple as a periodically driven pendulum or an autonomous conservative double pendulum can display chaotic dynamics. To identify the type of motion for a given system, one may look at a bundle of trajectories originating from a narrow cloud of points in phase space. The distance between any two such trajectories grows exponentially with time in the chaotic case; the growth rate is the so-called Lyapunov exponent. For regular motion, on the other hand, the distance in question may increase like a power of time but never exponentially; the corresponding Lyapunov exponent can thus be said to vanish. I should add that neither subexponential nor exponential separation can prevail indefinitely. Limits are set by Poincaré recurrences or, in some cases, the accessible volume of phase space. Usually, however, such limits become effective on time scales way beyond those on which regular or chaotic behavior is manifest.