From Regular to Chaotic Motions through the Work of Kolmogorov

Abstract

Since ancient times the problem of the stability of the solar system has been investigated by a wide range of scientists, from astronomers to mathematicians, to physicists. The early studies of P.S. Laplace, U. Leverrier, C.E. Delaunay and J.L. Lagrange were based on perturbation theory. Later H. Poincaré proved the non integrability of the three-body problem. It was only in the ‘50s that A. Kolmogorov provided a theorem which can be used in a constructive way to prove the stability of motions in nearly-integrable systems. A few years later, the pioneer work of Kolmogorov was extended by V.I. Arnold and J. Moser, providing the so-called KAM theory. Though the original estimates of the KAM theorem do not provide rigorous results in agreement with the astronomical predictions, the recent implementations of computer- assisted proofs show that KAM theory can be efficiently used in concrete applications. In particular, the stability of some asteroids (in the context of a simplified three-body problem) has been proved for the realistic values of the parameters, like the Jupiter-Sun mass ratio, the eccentricity and the semimajor axis.

KAM theory was the starting point for a broad investigation of the stability of nearly-integrable Hamiltonian systems. In particular, we review the theorems developed by N.N. Nekhoroshev, which allows to prove the stability on an open set of initial data for exponentially long times.

The numerical simulations performed by M. Henon and C. Heiles filled the gap between theory and experiments, opening a bridge toward the understanding of periodic, quasiperiodic and chaotic motions. In particular, the concept of chaos makes its appearance in a wide range of physical problems. The extent of chaotic motions is provided by the computation of Lyapunov’s exponents, which allow to measure the divergence of nearby trajectories. This concept was recently refined to investigate the behaviour of weakly chaotic motions, through the implemention of frequency analysis and Fast Lyapunov Indicators (FLI). We review the applications of such methods to investigate the topology of the phase space. Moreover, the computation of the FLI’s allowed to study the transition from Chirikov to Nekhoroshev regime and therefore it provides results about diffusion in Hamiltonian systems.