Lagrangian dynamics -- Noether's theorem -- Hamiltonian formulation -- Hamilton's principle function -- Hamilton's characteristic function -- Action-angle variables -- Abstract transformation theory -- Poisson brackets -- Liouville's theorem -- Preturbation theory -- The Henon-Heiles oscillator -- Discrete maps -- Lyapunov exponents -- The Poincar̐ưe-Birkoff theorem -- The KAM theorem -- Ergodic hypothesis -- Measure theory.
Summary:
"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 pathology. 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 Poincar̐ưe 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 Poincar̐ưe sections. This leads to the two great landmarks of chaos theory, the Poincar̐ưe-Birkoff 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"-- Provided by publisher.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.