The word "chaos" has two meanings that are almost exact opposites. In general usage, it means "wild unpredictable confusion". In physics and mathematics it often refers to the behavior of systems for which the tools of traditional mathematics fail and one is forced to think in a rigorous new semi-quantitative way.
The first person to realize this was the French mathematician Henri Poincaré. In the late 19th century he was challenged to prove that the solar system is stable, but instead raised the possibility that it might not be! If one ignores the gravitational attraction between the planets one is left with a solvable problem that appears in all undergraduate mechanics texts. Including these small perturbations leads to a problem that can be described with partial differential equations, but these equations have no proper solutions except possibly in terms of infinite series. These equations can be solved numerically of course, but the results show an odd combination of order and unpredictability. Poincaré went on to lay the foundation for the study of such systems. He identified criteria that make a system inescapably chaotic and showed how a kind of order can emerge from the chaos. In this century, the Russian mathematician Kolomogrov developed what has come to be called the KAM theorem; one of the crowning works of modern mathematics. He showed that there is an odd fractal-like landscape of infinite series solutions in otherwise unsolvable problems.
Lectures in Nonlinear Mechanics and Chaos Theory begins by reviewing the tools of traditional classical mechanics--the Hamiltonian formulation, abstract transformation theory, and perturbation theory--and shows how they ultimately fail. It then moves on to the landmarks of chaos theory, the Poincaré-Hopf or "hairy ball" theorem, followed by the Poincaré-Birkoff theorem for rational winding numbers, and finally, the KAM theorem. These are discussed in terms of rigorous mathematics and illustrated with numerous examples of computer-drawn solutions. It finishes with a discussion of the relevance of the KAM theorem and measure theory to the ergodic hypothesis.
This book is based on a one-quarter course in graduate mechanics that has been given in the Physics Department of Oregon State University. It is intended to be used as a textbook to review conventional mechanics and introduce students to more recent developments in chaos theory.
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