Preface

The main purpose of this book is to present and discuss, in an introductory and pedagogical way, a number of important recent developments in the dynamics of Hamiltonian systems of N degrees of freedom. This is a subject with a long and glorious history, which continues to be actively studied due to its many applications in a wide variety of scientific fields, the most important of them being classical mechanics, astronomy, optics, electromagnetism, solid state physics, quantum mechanics, and statistical mechanics. One could, of course, immediately point out the absence of biology, chemistry, or engineering from this list. And yet, even in such diverse areas, when the oscillations of mutually interacting elements arise, a Hamiltonian formulation can prove especially useful, as long as dissipation phenomena can be considered negligible. This situation occurs, for example, in weakly oscillating mechanical structures, lowresistance electrical circuits, energy transport processes in macromolecular models of motor proteins, or vibrating DNA double helical structures. Let us briefly review some basic facts about Hamiltonian dynamics, before proceeding to describe the contents of this book. The fundamental property of Hamiltonian systems is that they are derived from Hamilton’s Principle of Least Action and are intimately related to the conservation of volume, under time evolution in the phase space of their position and momentum variables qk , pk , k D 1; 2; : : : ; N , defined in the Euclidean phase space R . Their associated system of (first-order) differential equations of motion is obtained from a Hamiltonian function H , which depends on the phase space variables and perhaps also time. If H is explicitly time-independent, it represents a first integral of the motion expressing the conservation of total energy of the Hamiltonian system. The dynamics of this system is completely described by the solutions (trajectories or orbits) of Hamilton’s equations, which lie on a .2N 1/-dimensional manifold, the so-called energy surface, H.q1; : : : ; qN ; p1; : : : ; pN / D E. This constant energy manifold can be compact or not. If it is not, some orbits may escape to infinity, thus providing a suitable framework for studying many problems of interest to the dynamics of scattering phenomena. In the present book, however, we shall be exclusively concerned with the case where the constant energy manifold