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Matter in Motion

Brian Hayes

Structure and Interpretation of Classical Mechanics. Gerald Jay Sussman and Jack Wisdom with Meinhard E. Mayer. xxiv + 534 pp. The MIT Press, 2001. $65.

Seventeen years ago a book called Structure and Interpretation of Computer Programs swept through departments of computer science like a fresh breeze, bringing a new approach to teaching the fundamental ideas of the field. The authors--Harold Abelson and Gerald Jay Sussman with Julie Sussman--made those ideas seem at once intellectually serious and fun. It was a deep book, but also one that you could hand to an absolute beginner.

Structure and Interpretation of Classical Mechanics has much in common with the earlier volume, even beyond the deliberate similarity of title. Both books derive from courses taught at MIT; both rely heavily on the programming language Scheme; both have Gerald Jay Sussman among their authors. And this new work by Sussman and Jack Wisdom with Meinhard E. Mayer does much the same thing for classical mechanics that the earlier book did for computer science. The depth and the intellectual seriousness are there, and some of the fun. Regrettably, though, it's not a book for the complete beginner.

Classical mechanics is dominated by the towering figure of Newton, who had the unifying insight that motion in the heavens and motion on the earth can be understood through the same physical laws; Newton also devised the mathematical apparatus for describing that motion, namely the calculus. A drawing of Newton adorns the cover of Structure and Interpretation of Classical Mechanics, but inside the book he makes only a cameo appearance. Instead of Newtonian mechanics, Sussman and Wisdom focus on methods devised a century later by Euler and Lagrange, based on the calculus of variations. Newtonian mechanics starts with the forces acting on individual particles, then tracks the resulting accelerations, velocities and positions. The variational method operates at a higher level of abstraction, encoding the configuration of an entire system of particles in the coordinates of a single point in a higher-dimensional space, then following the path of that point as the physical configuration evolves. Given a starting and an ending configuration, the path between them is determined by "the principle of least action."

In Chapter 1 Sussman and Wisdom explain why the variational method is the better approach to studying matter in motion, and the rest of the book certainly validates that choice. Nevertheless, I can't help wishing for a Chapter 0 that would begin with the Newtonian formulation and progress to the Lagrangian, thereby recapitulating the historical development of these ideas. Such an introduction might have allowed timid readers to wade in from the shoreline rather than being thrown straightaway into deep water.

The chief innovation that Sussman and Wisdom bring to this subject is their emphasis on computation. But don't look for a CD-ROM in the back of the book with animations of orbiting planets and swinging pendulums; in these pages computation is not a multimedia supplement but rather is woven directly into the narrative. Many ideas that might otherwise be expressed in mathematical form are given as procedures in Scheme (a dialect in the Lisp family of programming languages). The version of Scheme employed in the book is greatly augmented by a package called Scmutils, which essentially turns the language into a system for symbolic and numerical mathematics, a little like Maple or Mathematica. (Both Scmutils and the underlying implementation of Scheme—as well as the entire book itself—are available via Sussman's Web site at

Not all of the mathematics takes the form of Scheme programs. There is also much that appears on first glance to be written in conventional notation. On closer examination, however, the equations turn out to have a slightly unusual form. They follow a carefully restricted syntax and semantics, allowing a one-to-one mapping between mathematical and Scheme expressions.

One aim of writing in such a computer-parsable language is to cleanse the notation of ambiguities. Sussman and Wisdom report that they found the task harder than they expected, but illuminating. "We quickly learned that many things we thought we understood we did not in fact understand. Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning."

There is also a larger purpose for the focus on computation: Because the mathematics is "executable," the examples and problems addressed need not be limited to the few tidy cases that can be worked out by hand. And indeed a major theme of the book is the exploration of chaotic motion, where exact analytic results are scarce but computational methods work well. For example, they ask why the Earth's moon rotates stably, whereas Saturn's satellite Hyperion tumbles chaotically. The qualitative difference between the two systems is not clear until you follow the motion for many orbital periods, which would be extremely arduous without computer assistance.

Classical mechanics is often taken as a model of rigor and exactness in the natural sciences, and it's an easy step from that austere view of the field to seeing it as a closed and finished body of knowledge. The very term "classical" tends to suggest that everything is known and nothing remains to be discovered. A reading of Sussman and Wisdom will leave you with a very different impression.-Brian Hayes

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