Essentials of Hamiltonian Dynamics, John H. Lowenstein, Cambridge U. Press, New York, 2012. $60.00 (188 pp.). ISBN 978-1-107-00520-4
If two people are teaching separate courses on Hamiltonian dynamics, there is a good chance that the classes will be completely different. A course for mathematicians will be quite different from one for physicists; an introductory course will be different from one for advanced students. Also, the instructor’s selection of topics will usually reflect his or her personal taste.
The choice of a textbook certainly depends on a course’s aim and style. For a course aimed at students who intend to study the internal structure of Hamiltonian dynamics rather than to use it for applications, one good option is Vladimir Arnold’s Mathematical Methods of Classical Mechanics (2nd edition, Springer, 2010). For students looking for tools to attack problems in celestial mechanics, there’s the useful text by Kenneth Meyer, Glen Hall, and Dan Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (2nd edition, Springer, 2010).
John Lowenstein’s Essentials of Hamiltonian Dynamics will be particularly helpful for physics instructors. The ideal reader for this book is a graduate physics student who has already been exposed to the Lagrangian and Hamiltonian formulations of classical mechanics, has a basic knowledge of the theory of dynamical systems, and now needs sufficient familiarity with Hamiltonian dynamics to apply its methods and results to specific models and systems.
The introductory chapter is a review of the fundamentals of classical mechanics; it introduces the notion of configuration space as a differential manifold, briefly covers the Lagrangian and Hamiltonian formulations, and works through some basic examples. Chapter 2 introduces the Hamiltonian formalism. Chapter 3 defines integrable systems; considers several examples, including the spherical pendulum and the three-particle Toda model; and formulates the Liouville–Arnold theorem and gives an idea of its proof. The basics of canonical perturbation theory are described in chapter 4. Then in chapter 5, the author addresses order and chaos in Hamiltonian systems and introduces such topics as the Kolmogorov-Arnold-Moser (KAM) theory, Lyapunov exponents, and several illustrative systems, including the kicked oscillators and a chaotically rotating moon. He devotes chapter 6, the final chapter, to the dynamics of the swing–spring model that serves as an example of how all the previously introduced notions and techniques can be used to study the properties of a particular system. Each chapter concludes with a collection of exercises.
A special feature of the book that may attract some readers, and repel others, is the extensive use of the Mathematica computational software. Mathematica helped create pictures and numerical data in the text, and many exercises are formulated as numerical experiments that students are expected to solve using Mathematica. Also, the author provides an appendix that includes samples of the Mathematica programs used in the text.
Overall, Essentials of Hamiltonian Dynamics offers a minimalistic approach, which, depending on the reader, can be a plus or a minus. The choice of topics is natural for a one-semester graduate course, but many traditional “enrichment topics” are mentioned barely or not at all. In general, the book mostly emphasizes how this or that notion or phenomenon appears in particular models. Also, the exposition of the standard topics is sometimes sketchy. For example, the entire KAM theorem, which addresses quasi-periodic motions in near-integrable systems, is presented in only five pages. The author also uses a bare minimum of mathematical tools. For instance, although the notion of differential manifold was introduced and used, the book completely avoids the language of differential forms. The minimalistic approach also reveals itself in the references section, which contains a mere 47 items, mostly to original sources rather than to easier-to-read modern expositions.
This book does not provide a comprehensive exposition on the state of the art in Hamiltonian dynamics. Instead, it provides a shortcut to many amazing and useful applications of this fascinating subject. I have no doubt that many instructors and students will benefit from this text.
