Structure and Interpretation of Classical Mechanics , Gerald Jay Sussman and Jack Wisdom , with M. E. Mayer MIT Press, Cambridge, Mass., 2001. $60.00 (534 pp.). ISBN 0-262-019455-4
The evolution of classical mechanics over the last century or so has been marked by two watersheds. The first was Henri Poincaré’s discovery that studying the geometry of trajectories in phase space offered far more insight into most problems in mechanics than did the search for exact solutions or perturbation expansions. The second was the emergence of fast digital computers, which enabled numerical experiments that put flesh on the bones of the mathematical theorems and proved their relevance to real problems.
Remarkably, as we enter a new century, these discoveries are only beginning to trickle into the physics curriculum. Many or most senior undergraduate- and even graduate-level mechanics courses still focus on the 18th- and 19th-century discoveries of Leonhard Euler, Joseph-Louis Lagrange, Pierre Simon de Laplace, William R. Hamilton, and others, as expounded in vintage texts such as Lev Landau and E. M. Lifshitz’s Mechanics 3rd edition (Pergamon, 1989), Herbert Goldstein’s Classical Mechanics 2nd edition (Addison-Wesley, 1980), and Jerry B. Marion and Stephen T. Thornton’s Classical Dynamics (Saunders, 1995).
Part of the reason for this obsolescent curriculum is the shortage of suitable physics textbooks that describe 20th-century developments in mechanics. For example, Vladimir I. Arnold’s Mathematical Methods of Classical Mechanics (Springer, 1989) is too mathematical for most physics students; Allen J. Lichtenberg and M. A. Lieberman’s Regular and Chaotic Dynamics (Springer, 1992) is too specialized for a general course. Jorge V. José and Eugene J. Saletan’s Classical Dynamics (Cambridge, 1998) is one of the very few books designed for general courses in advanced mechanics that includes more than a token chapter on chaos and other developments since Hamilton.
Gerald J. Sussman and Jack Wisdom’s Structure and Interpretation of Classical Mechanics is a notable and innovative entry into this fallow field. Wisdom, a professor of planetary science at MIT, has led the application of modern methods to solar-system dynamics, with spectacular successes in resolving long-standing problems such as the formation of the Kirkwood gaps in the asteroid belt and the long-term stability of the solar system. Sussman is a professor of electrical engineering at MIT, the author (with Harold Abelson) of a widely used introductory computer science text (Structure and Interpretation of Computer Programs, MIT Press, 1996), and the designer of a successful special-purpose computer for long solar-system integrations.
Sussman and Wisdom’s book covers most of the standard topics of 19th-century mechanics: Lagrangians, Hamiltonians, rigid-body motion, canonical transformations, rotating coordinates, action-angle variables, and so on. Equally important, and much more unusual, is its broad coverage of the major concepts of 20th-century mechanics: chaos, homoclinic tangles, invariant curves, Liapunov exponents, Lie transforms, perturbation theory, the standard map, surfaces of section, and so forth. The topics are well selected, and their implications and subtleties are thoroughly and carefully discussed. In contrast to most other writers on this subject, the authors use the tools they develop in their research, and it shows.
A particularly vexing problem for students of classical mechanics is that traditional notations are often ambiguous and context-dependent, especially when applied to variational principles. Sussman and Wisdom pay careful attention to this problem. They use a modern and carefully defined notation and, more radically, present many of their formulas as procedures in the computer language Scheme (a dialect of LISP), with a close correspondence between their mathematical notation and the grammar of the computing language. (All of the required software is freely available on the Web.) They argue, quite plausibly, that the discipline required to express the notions of classical mechanics as computational objects is an important learning tool.
All textbooks have defects, and one as original as this is bound to be less polished than its successors. I was surprised at the absence of several major topics, such as adiabatic invariance, dissipative systems, and normal modes. It is unfortunate that the book almost ignores numerical integration methods; not only is numerical integration indispensable for understanding nonintegrable systems, but symplectic integration methods provide one of the best examples of the role of geometry in determining the behavior of dynamical systems. I found the book difficult to browse through, perhaps because the lengthy Scheme expressions can be distracting, as are the more than 200 footnotes. The book is not well-designed as a reference work, having only a short bibliography and other frustrating features (For example, a formula for the important Baker-Campbell-Hausdorff series for combining exponentials of noncommuting operators gives only the first nontrivial term!).
I would grade the book A+ for originality and authority, A for its choice of topics, and B for clarity and ease of use. Despite its minor flaws, the book will almost certainly change the way its readers think about mechanics, and, more important, it may well change the way they teach the subject.
