Complex Hamiltonian Dynamics
This book introduces and explores modern developments in the well established field of Hamiltonian dynamical systems. It focuses on high degree-of-freedom systems and the transitional regimes between regular and chaotic motion. The role of nonlinear norma
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Introduction
Abstract Chapter 1 starts by defining a dynamical system in terms of ordinary differential equations and presents the fundamental framework within which one can study the stability of their equilibrium (or fixed) points, as developed by the great Russian mathematician A. M. Lyapunov. The concept of Lyapunov Characteristic Exponents is introduced and two theorems by Lyapunov are discussed, which establish criteria for the asymptotic stability of a fixed point. The mathematical setting of a Hamiltonian system is presented and a third theorem by Lyapunov is stated concerning the continuation of the linear normal modes of N harmonic oscillators, when the system is perturbed by adding to the Hamiltonian nonlinear terms higher than quadratic. Finally, we discuss the meaning of complexity in Hamiltonian dynamics, by referring to certain weakly chaotic orbits, which form complicated quasi-stationary states that are well-approximated by the principles of nonextensive statistical mechanics for very long times.
1.1 Preamble The title of this book consists of three words and it is important to understand all of them. We shall not start, however, with the first one, as it is the most difficult to define. It may also turn out to be the most fascinating and intriguing one in the end. Rather, we will start with the last word: What is dynamics? Well, it comes from the Greek word “dynamis” meaning “force”, so it will be very natural to think of it in the Newtonian context of mechanics as describing the effect of a vectorial quantity proportional to the mass and acceleration of a given body. But that is a very limited interpretation. Dynamics, in fact, refers to any physical observable that is in motion with respect to a stationary frame of reference. And as every student knows this motion may very well be uniform (i.e. with constant velocity) and hence not involve any force at all. Thus we will speak of dynamics when we are interested to know how the state of an observable changes in time. This may represent, for example, the position or T. Bountis and H. Skokos, Complex Hamiltonian Dynamics, Springer Series in Synergetics, DOI 10.1007/978-3-642-27305-6 1, © Springer-Verlag Berlin Heidelberg 2012
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1 Introduction
velocity of a mass particle in the three-dimensional space, the flow of current in an electrical circuit, the concentration of a chemical substance, or the population of a biological species. When we refer to a collection of individual components, we shall speak of the dynamics of a system. Most frequently, these components will interact with each other and act interdependently. And this is when matters start to get complicated. What is the nature of this interdependence and how does it affect the dynamics? Does it always lead to behavior that we call “unpredictable”, or chaotic (in the sense of extremely sensitive dependence on initial conditions), or does it also give rise to motions that we may refer to as “regular”, “ordered” or “predictable”? And what about the intermediate regime between order an
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