Theory of the Liapunov exponents of Hamiltonian systems and a numerical study on the transition from regular to irregular classical motion

Abstract

The theory of Liapunov exponents of Hamiltonian systems is reviewed and the connection between vanishing Liapunov exponents and constants of motion is discussed. There are at least 2 N vanishing Liapunov exponents if there are N independent constants of motion. The Liapunov exponents are then used to determine the relative weight of the irregular motion on the energy shell as well as the Kolmogorov–Sinai entropy for two model systems. The trajectory and the stability matrix from which the Liapunov exponent is deduced are integrated with the aid of a Taylor-expansion integration scheme.