Lectures on Nonlinear Mechanics and Chaos Theory

Albert W Stetz

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813141377
• Topic: Physical Sciences
• Subtopic: Mechanics

This elegant book presents a rigorous introduction to the theory of nonlinear mechanics and chaos. It turns out that many simple mechanical systems suffer from a peculiar malady. They are deterministic in the sense that their motion can be described with partial differential equations, but these equations have no proper solutions and the behavior they describe can be wildly unpredictable. This is implicit in Newtonian physics, and although it was analyzed in the pioneering work of Poincaré in the 19th century, its full significance has only been realized since the advent of modern computing. This book follows this development in the context of classical mechanics as it is usually taught in most graduate programs in physics. It starts with the seminal work of Laplace, Hamilton, and Liouville in the early 19th century and shows how their formulation of mechanics inevitably leads to systems that cannot be "solved" in the usual sense of the word. It then discusses perturbation theory which, rather than providing approximate solutions, fails catastrophically due to the problem of small denominators. It then goes on to describe chaotic motion using the tools of discrete maps and Poincaré sections. This leads to the two great landmarks of chaos theory, the Poincaré–Birkhoff theorem and the so-called KAM theorem, one of the signal results in modern mathematics. The book concludes with an appendix discussing the relevance of the KAM theorem to the ergodic hypothesis and the second law of thermodynamics.

Lectures on Nonlinear Mechanics and Chaos Theory is written in the easy conversational style of a great teacher. It features numerous computer-drawn figures illustrating the behavior of nonlinear systems. It also contains homework exercises and a selection of more detailed computational projects. The book will be valuable to students and faculty in physics, mathematics, and engineering.

Contents:

• Lagrangian Dynamics
• Canonical Transformations
• Abstract Transformation Theory
• Canonical Perturbation Theory
• Introduction to Chaos
• Computational Projects

Readership: Students who want to learn the subject of nonlinear mechanics and chaos theory from first principles.
Key Features:

• It presents chaos theory as a logical consequence of the traditional formulation of classical mechanics
• It is fine-tuned to the requirements of a graduate course in physics
• It contains numerous computer-drawn figures illustrating the behavior of chaotic systems, figures that are not available in any other book

Chapter 1

Lagrangian dynamics

1.1Introduction

The possibility that deterministic mechanical systems could exhibit the behavior we now call chaos was first realized by the French mathematician Henri Poincaré sometime toward the end of the nineteenth century. His discovery emerged from analytic or classical mechanics, which is still part of the foundation of physics. To oversimplify a bit, classical mechanics deals with those problems that can be “solved,” in the sense that it is possible to derive equations of motions that describe the positions of the various parts of a system as functions of time using standard analytic functions. Nonlinear dynamics treats problems that cannot be so solved, and it is only in these problems that chaos can appear. The simple pendulum makes a good example. The differential equation of motion is

The sin θ is a nonlinear function of θ. If we linearize by setting sin θ ≈ θ, the solutions are elementary functions, sin ωt and cos ωt. If we keep the sin θ, the solutions can only be expressed in terms of elliptic integrals. This is not a chaotic system, because there is only one degree of freedom, but if we hang one pendulum from the end of another, the equations of motion are hopeless to find (even with elliptic integrals) and the resulting motion can be chaotic.¹

In order to arrive at Poincaré’s moment of discovery, we will have to review the development of classical mechanics through the nineteenth century. This material is found in many standard texts, but I will cover it here in some detail. This is partly to insure uniform notation throughout these lectures and partly to focus on those things that lead directly to chaos in nonlinear systems. We will begin formulating mechanics in terms of generalized coordinates and the Lagrange equations. We then study Legendre transformations and use them to derive Hamilton’s equations of motion. These equations are particularly suited to conservative systems in which the Hamiltonian is constant in time, and it is such systems that will be our primary concern. It turns out that Legendre transformations can be used to transform Hamiltonians in a myriad of ways. One particularly elegant form uses action-angle variables to transform a certain class of problems into a set of uncoupled harmonic oscillators. Systems that can be so transformed are said to be integrable, which is to say that they can be “solved,” at least in principle. What happens, Poincaré asked, to a system that is almost but not quite integrable? The answer entails perturbation theory and leads to the disastrous problem of small divisors. This is the path that led originally to the discovery of chaos, and it is the one we will pursue here.

1.2Generalized coordinates and the Lagrangian

Vector equations, like F = ma, seem to imply a coordinate system. Beginning students learn to use cartesian coordinates and then learn that this is not always the best choice. If the system has cylindrical symmetry, for example, it is best to use cylindrical coordinates: it makes the problem easier. By “symmetry” we mean that the number of degrees of freedom of the system is less that the dimensionality of the space in which it is imbedded. The familiar example of the block sliding down the incline plane will make this clear. Let’s say that it’s a two dimensional problem with an x-y coordinate system. The block is constrained t...