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Mathematical Methods of Classical Physics

Vicente Cortés and Alexander S. Haupt
Publication Date: 
Number of Pages: 
Springer Briefs in Physics
[Reviewed by
Warren Johnson
, on

This is an attractive little volume whose scope is much narrower than its title suggests. For regular readers of this column, it is a Michael Berg kind of book. The title must have put him off.

The authors teach in the Department of Mathematics and Center for Mathematical Physics at the University of Hamburg in Germany, and the book grew out of a graduate course offered there in winter 2015/2016. Their English is fine, if occasionally German-inflected, e.g. (p. 3) “Jet bundles do not receive widespread attention in the physics literature up to now.” Chapter 1 is a brief introduction. The authors adopt there a curious definition of classical physics: quantum mechanics doesn’t count, but relativity does. They state Newton’s laws of mechanics and prepare the way for the formulations of Lagrange and Hamilton.

Cortés and Haupt contrast their book (hereafter CH) with several classics: Foundations of Mechanics, by Abraham and Marsden; Mathematical Methods of Classical Mechanics, by Arnold; and Goldstein’s Classical Mechanics, which I used in graduate school. The first two editions of Goldstein were the standard textbook for the graduate physics class of the same name for many years, going back to the early 1950s; a third edition, revised by Poole and Safko, is still in use today. Fans of catty reviews might enjoy Clifford Truesdell’s take on the first edition, apparently too negative to be printed by the journal that had solicited it, on pp. 144–147 of his An Idiot’s Fugitive Essays on Science.

Cortés and Haupt say that these books “require a strong background in physics and do not put strong emphasis on mathematical rigor”, which is certainly true of Goldstein, though it might be a little rough on the others. They see CH as “complementary” in that it “takes a different, more mathematical, perspective at these central topics of classical physics. It puts emphasis on a mathematically precise formulation of the topics while conveying the underlying geometrical ideas.” It is “primarily directed at readers with a background in mathematical physics and mathematics,” but there is no specific statement of what the reader is supposed to know.

So it’s a little clunky sometimes, but still one turns to the chapters on Lagrangian and Hamiltonian mechanics with anticipation. The beauty of this subject, intimately bound up with the fundamental ideas of the calculus of variations, nearly convinced your reviewer that he was really a physicist many years ago, in spite of a mountain of evidence to the contrary. Definition 2.1 shows the reader what she is in for: “A Lagrangian mechanical system is a pair \((M,\mathcal{L})\) consisting of a smooth manifold \(M\) and a smooth function \(\mathcal{L}\) on \(TM\).” There is a second sentence, but it does not explain what \(TM\) is. For that one is expected to turn to a list of symbols on pp. ix–xi, where one finds that \(TM\) stands for the tangent bundle of \(M\). Nowhere is there a definition of tangent bundle.

If you don’t mind this kind of thing (and there are many other unexplained terms in chapter 2, such as pseudo-Riemannian manifold and vertical lift and Killing vector field), then this chapter is a nice brisk survey of Lagrangian mechanics, including a preliminary version of Noether’s theorem. Section 2.3 on motion in a radial potential, including a derivation of Kepler’s laws, is well done and easier to follow.

Chapter 3 on Hamiltonian mechanics begins by (tersely) defining a symplectic manifold. The claim in the introduction that “all theorems are proved” is violated at the top of p. 20, where a fundamental result of Darboux on differential forms is merely quoted from Abraham and Marsden. The beautiful Hamilton equations are in Proposition 3.7, and the relation between the Euler-Lagrange equations and Hamilton’s is elucidated in section 3.2. Section 3.3 observes that both sets of equations are usually nonlinear and discusses linear approximation, concluding with the standard problem of two coupled harmonic oscillators.

Chapter 4 continues to develop Hamilton’s theory, culminating in the Hamilton-Jacobi equation that in some sense compresses the whole theory into a single formula. For this reader, the beauty of the idea is considerably disfigured by the high level of the treatment. Except for the introduction, this is easily the shortest of the five chapters, and like the two previous chapters it concludes with a physical example, the third and last in the book.

Chapter 5 on classical field theory is much the longest and appears to be the real point of CH. While I did not enjoy the authors’ approach to the Lagrange/Hamilton theory, they have presented it in such a way that this material is a natural continuation. Jets and jet bundles come in section 5.1. Noether’s theorem on conservation laws is proved in section 5.2 and further illuminated in section 5.3. Section 5.4 treats Yang-Mills theory, with Maxwell’s equations derived as a special case in Example 5.32, which perhaps I should count as a fourth physical example.

The ideal reader of CH would have a strong background in geometry and a strong desire to learn the material of chapter 5, neither of which, unfortunately, applies to your reviewer. I doubt there is another book that does this as well in as little space. For a mathematical reader wanting to learn the classical Lagrange/Hamilton theory, the best source is probably Lanczos’s The Variational Principles of Mechanics, although it is a little windy. Chapter 4 of Gelfand and Fomin’s Calculus of Variations is another good treatment. The reader with a serious interest in history (CH has none) and good German might also consider Jacobi’s classic Vorlesungen über Dynamik.

Warren Johnson ([email protected]) is Associate Professor of Mathematics at Connecticut College. He is trying to finish a book on \(q\)-analysis.

See the table of contents in the publisher's webpage.