2006, Vol.9, No.3, pp.209-239

In this paper I discuss some aspects of a problem with a long history, which continues to be of current interest due to its great importance to many applications: The stability of motion in N-degree of freedom Hamiltonian systems. I will start with N small and proceed to the case of N arbitrarily large, in an attempt to understand the thermodynamic limit, where and statistical mechanics is expected to take over from classical mechanics. Our domain is the Euclidian phase space of the q_k, p_k, k=1,2,...,N, position and momentum coordinates. The dynamics is governed by Hamilton's equations of motion and the solutions (or orbits) lie on a (2N-1)-dimensional compact manifold (the so-called "energy surface"), defined by H(q₁, ..., q_N, p₁, ..., p_N)=E, where H is the Hamiltonian and E is the (constant) energy of the system. Of primary importance in our discussion is the connection between local and global dynamics, i.e. the seemingly paradoxical relevance of events occurring in small scale regions of the energy surface, to the stability of motion in large domains, affecting the properties of the system as a whole. This link is provided by a detailed study of what I call Simple Periodic Orbits (SPOs), i.e periodic solutions where all variables oscillate with equal frequencies, , returning to the same values after a single maximum (and minimum) in their evolution over one period T.

We will start by recalling some fundamental concepts concerning the stability of dynamical systems, as introduced by one of the forefathers of this field, the great Russian Mathematician A. M. Lyapunov, more than 110 years ago. First, we shall review his two main methods for studying the solutions in the vicinity of equilibrium points that led to his proof of the existence of periodic solutions, as continuations of the corresponding oscillations of the linearized system of equations. We will then apply his theory of the continuation of normal modes of N-degree of freedom Hamiltonian systems to the famous Fermi-Pasta-Ulam lattice to explain how these SPOs can help resolve the paradox of the FPU recurrences. I will then discuss how the study of other SPOs, corresponding to in-phase and out-of-phase oscillations in the FPU and other Hamiltonians, help us understand the transition to large scale chaotic behavior, characterized by invariant spectra of Lyapunov exponents, as well as identify domains where the motion is still quasiperiodic, lying on invariant N-dimensional tori (on which the are rationally independent). Finally, I will report on a recent discovery of a very efficient spectrum of indices distinguishing ordered from chaotic motions in conservative dynamical systems, called the GALI_k, k=1,2,...,2N. These represent an important generalization of the SALI, used by many researchers to identify domains of chaos and order not only in N-degree-of-freedom Hamiltonian systems, but also 2N-dimensional symplectic maps.

Full text:  Acrobat PDF  (766KB)  

Copyright © Nonlinear Phenomena in Complex Systems. Last updated: September 6, 2006