By Albert W Stetz
World Scientific
This concise book provides a rigorous introduction to the theory of nonlinear mechanics and chaos, suitable for students across physics, mathematics and engineering.
Nonlinear dynamics treats problems that cannot be “solved”, in the sense that it is not possible to derive equations of motion that describe the positions of the various parts of a system as functions of time using standard analytic functions. If, on one side, the formulations of mechanics of Lagrange and Hamilton lead to systems that cannot be solved in the usual sense of the word, perturbation theory, in turn, fails in providing approximate solutions because of the problem of small dividers. This is the path that led originally to the discovery of chaos, and it is the one that the author pursues in the book.
The first part is dedicated to the basic concepts of the Lagrangian and Hamiltonian formulation of mechanics, and to canonical transformations. The author then deals with more advanced topics, including Liouville’s theorem and perturbation theory. In the third part of the book, the modern theory of chaos is introduced. The author describes chaotic motion using the tools of discrete maps and Poincaré sections, along with the Poincaré–Birkhoff and Kolmogorov–Arnold–Moser (KAM) theorems and their applications.
Each chapter is accompanied by a set of problems, with the last section providing more advanced projects that require some expertise in computing. As a conclusion, an appendix discusses the relevance of the KAM theorem to the ergodic hypothesis and the second law of thermodynamics.
