Foundations of Physics 32,323-326 (2002)

Book Review of Structure and Interpretation of Classical Mechanics
by G. J. Sussman and J. Wisdom, with Hardy Mayer (M.I.T. Press)

Piet Hut
Institute for Advanced Study
Princeton, NJ

This is the first book I have come across that explains classical mechanics using the variational principle in such a way that there are no ambiguities in either the presentation or the notation. The resultant leap toward clarity and precision is likely to influence a new generation of physics students, opening their eyes to the beauty of classical mechanics. Quite likely, some of these students will be inspired to find newer and deeper interpretations of classical mechanics. In turn, such deeper insight may well lead to a deeper understanding of quantum mechanics. After all, classical mechanics is only a limiting case of quantum mechanics, and the variational principle is the main bridge between the two.

Many physics students are introduced to Hamiltonians and Lagrangians only in their first course on quantum mechanics. It often comes as a surprise to them to hear that both Hamilton and Lagrange were long dead by 1925, and that in fact their methods had been around for most of the nineteenth century. Rushing on to quantum mechanics this way turns the historical development on its head, thus obscuring the most fundamental relationship between classical and quantum mechanics, and making the latter seem even more mysterious and weird than it really is. As a result, students come away with the impression that quantum mechanics uses tools that are fundamentally different from those used in classical mechanics. Furthermore, it seems as if one has to go through great trouble to extract something resembling classical equations of motion from approximate treatments of expectation values of quantum mechanical variables.

I wish I had been introduced to classical mechanics through this book before learning the details of quantum mechanics. As it happened, I was born thirty years too early to read this book as a freshman. Instead, I had to stumble my way through various mechanics representations, in a way that is not atypical of what happened to many of my colleagues.

At the end of my second year as an undergraduate at the University of Utrecht, in The Netherlands, I had completed a detailed course in classical mechanics and an introductory course, more something like a guided tour, in quantum mechanics. I knew that after the summer I would get my first real course in quantum mechanics, and I very much looked forward to it. From the introductory tour I had realized that the world of the quantum offered a totally different reality than the clockwork world of classical mechanics. I was very curious to see how the twain would meet: how it could be possible that the intrinsically spontaneous, unruly and unreproducible quantum world could ever have given rise on larger scales to the staid semblance of the clockwork world of classical mechanics.

Just around this time, an older student told me that I first might want to read about the variational principle in classical mechanics. He gave me a bound volume of course notes on Lagrangian and Hamiltonian dynamics. I vividly remember taking the volume under my arm, walking to a nice terrace next to one of the Utrecht canals, and getting engrossed in the text. The applications were all very familiar, only the approach was sheer magic. I saw in classical garb the equivalence of wave functions and phase interference in classical mechanics, in ways that had been discovered more than a century before quantum mechanics! How was that possible? What did that mean? I still remember putting the volume down, after a couple hours, and staring at the way the sunlight, reflected from the waves on the water, was painting ever-shifting caustics inside the arches of the stone bridge next to my terrace. No better metaphor could have presented itself in front of my eyes. So this was how the world is hanging together, I thought, through waves and phases and interference, even on the classical level!

For several days I felt the impact of this revelation. Now I really wanted to know what was going on. Soon I went through the library in search of books on the variational principle in classical mechanics. I found several heavy tomes, borrowed them all, and started on the one that looked most attractive. Alas, it didn't take long for me to realize that there was quite a bit of hand-waving involved. There was no clear definition of the procedure used for computing path integrals, let alone for the operations of differentiating them in various ways, by using partial derivatives and/or using an ordinary derivative along a particular path. And when and why the end points of the various paths had to be considered fixed or open to variation also was unclear, contributing to the overall confusion.

Working through the canned exercises was not very difficult, and from an instrumental point of view, my book was quite clear, as long as the reader would stick to simple examples. But the ambiguity of the presentation frustrated me, and I started scanning through other, even more detailed books. Alas, nowhere did I find the clarity that I desired, and after a few months I simply gave up. Like generations of students before me, I reluctantly accepted the dictum that `you should not try to understand quantum mechanics, since that will lead you astray for doing physics', and going even further, I also gave up trying to really understand classical mechanics! Psychological defense mechanisms turned my bitter sense of disappointment into a dull sense of disenchantment.

Given this background, I was delighted to read the preface of Structure and Interpretation of Classical Mechanics. The very first quote from Jacobi, cited by Arnold, showed that the authors must have had similar struggles as I did as a student: ``In almost all textbooks, even the best, this [variational] principle is presented so that it is impossible to understand.'' To see a book that sets out on the very first page to attack this problem is heart-warming.

I hope that a new generation of students will stumble upon Structure and Interpretation of Classical Mechanics, and thereby will make a better and deeper acquaintance with both classical and quantum mechanics than I did. The central feature of this new book is that it has a built-in guarantee that there is zero ambiguity in any of the 1200 or so mathematical equations that appear. This sounds like a remarkable if not impossible claim. The key here is that every mathematical expression is in one-to-one correspondence with an equivalent expression written in computer code. Each of these expressions has been compiled and tested out by the authors and by many of their students over several years, while the authors were developing and simultaneously teaching the material in this book.

The key to using a computer language is summarized by the authors, also in the preface: ``The requirement that the computer be able to interpret any expression provides strict and immediate feedback as to whether the expression is correctly formulated. Experience demonstrates that interaction with the computer in this way uncovers and corrects many deficiencies in understanding.''

It may seem strange that a piece of computer code can capture such notions as evaluating various types of variations of functions containing partial derivatives along either arbitrary or prescribed paths in phase space. When expressed in more traditional programming languages encountered in physics or in business, such as Fortran or C or C++, writing a package to model variational mechanics would have been a very complex task, resulting in a much more opaque product. Fortunately, the authors chose a functional language, Scheme, a particularly lean Lisp-like language, in which these ideas can be expressed in a very economical and clear way. Functional composition, the key to mathematically clarifying variational mechanics, has an exact counterpart in the functional programming approach that is central in Scheme.

Scheme is widely available, and often for free as open-source shareware, and there are some excellent textbooks on Scheme, notably "Structure and Interpretation of Computer Programs", by H. Abelson and G. J. Sussman (M.I.T. Press, 2nd edition, 1996). While such books will be helpful, the current book is self-contained: an appendix gives an elementary introduction to Scheme, which suffices for the book.

One characteristic of Scheme is that it is easy to learn: the syntax of the language is far simpler than that of, say, Fortran or C (to say nothing of C++). Even so, if the reader is curious about the full power of Scheme as well as the principles behind its design, the Abelson/Sussman book is well worth reading. The invitation to both abstract and compact thinking offered by Scheme leaves its mark, once you start playing with it, even for a short time. In my case, as soon as I read the first edition in 1985, I was hooked on Scheme as my language of choice. And even though I find myself forced to use other languages in collaborative projects in astrophysics, my programming style in those other languages, too, clearly reveals the inspiration I have received from Scheme.

The content of the book is thoroughly modern, and covers topics from Lie series to a variety of topics in chaos theory. The authors have an impressive track record in the latter area: they were the first ones to show the fact that the solar system is chaotic, a rather startling discovery when it was announced in 1988 (in Science, 241, 433; using a special-purpose computer, the Digital Orrery, they found that Pluto's orbit has an inverse Lyapunov exponent of 20 Myr). Finally, all chapters in the book are interspersed with numerous helpful exercises, all of which have been tested extensively during the six years that the authors have taught the material presented in this book to undergraduates at M.I.T.

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