Thereâ€™s something almost magical about the phrase, â€œThe Calculus of Variations,â€ conjuring up tales of titans ranging from Isaac Newton to David Hilbert. Everyone surely recalls the story of the least-time path in a gravitational field, the brachistochrone, posed by Johann Bernoulli and solved by Isaac Newton. Almost all of us were likely exposed to it first from Eric Temple Bellâ€™s universally beloved Men of Mathematics
: â€œIn1696 Johann Bernoulli and Leibniz between them concocted two devilish challenges to the mathematicians of Europe. The first is still of importance; the second is not in the same class â€¦ [The first problem] is the problem of the brachistochrone
... After the problem had baffled the mathematicians of Europe for six months, it was proposed again, and Newton heard of it for the first time on January 29, 1696 [1697?], when a friend communicated it to him. He had just come home, tired out, from a long day at the Mint. After dinner he solved the problem (and the second as well), and the following day communicated his solutions to the Royal Society anonymously. But for all his caution he could not conceal his identity â€¦ On seeing the solution Bernoulli at once exclaimed, â€˜Ah! I recognize the lion by his paw.â€™â€ [loc. cit., p. 115] Taking into account the late Dr. Bellâ€™s well-known love of a good tale, touching even upon occasional embellishment, we should note that, in fact, the brachistochrone problem actually goes back to Galileo, in 1638, and was in fact solved by both Johann and Jacob Bernoulli, as well as by Leibniz, in addition to the English Lion himself. This we learn from Dacorogna on p.1 of his Introduction to the Calculus of Variations
, as well as a lot of other marvelous historical stuff. And, sure enough, he soon makes his way to the Dirichlet problem and closes his brief historical introduction on p.2, with the observation that Problems 19, 20, and 23 of Hilbertâ€™s list of the renowned 23 â€œParis Problemsâ€ concern the calculus of variations. What a pedigree!
So it is, then, that this wonderful book is imbued with a marvelous historical perspective so that the reader is taught some very beautiful mathematics fitted in the proper historical perspective. Moreover, as regards mathematics per se, the book is full of terrific hard (as opposed to soft) analysis focused on a general theme that is exemplified by the authorâ€™s astute and elegant choice of topics. Dacorognaâ€™s arsenal (amassed in Chapter 1) includes Hölder continuity, Sobolev spaces and convex analysis, in addition to more familiar fare; thereafter he goes after the following big game: the Euler-Lagrange and Hamilton-Jacobi equations, the Dirichlet problem, of course, and then minimal surfaces and the isoperimetric inequality.
Dacorognaâ€™s dominant focus on the Dirichlet problem is explicit from the very outset: already on p.2 he characterizes the problem as â€œthe most celebrated problem of the calculus of variationsâ€ and points towards Hilbertâ€™s famous solution of the problem (recounted so beautifully in Constance Reidâ€™s Hilbert). Lebesgue and Tonelli soon joined in and, collectively, their approaches sired what are today called â€œthe direct methods of the calculus of variations.â€ Wasting no time, on p.3 Dacorogna presents a very accessible model for the Dirichlet problem, serving as a prototype and more, consisting in minimizing the appropriate Dirichlet integral as a function of â€œadmissibleâ€ functions whence everything largely depends on what one posits as the indicated a priori hypotheses on the admissible functions (in this connection see p.79 for example). The model given on p.3 sets the tone (and the stage) for much of what follows and evolves magnificently through the ensuing pages, lending the book an impressive cohesion.
This having been said, it should be noted that while Dacorogna advertises his book as â€œa concise and broad introductionâ€ to the calculus of variations at an undergraduate and beginning graduate level, he does presuppose the reader to be able and willing to work hard and do battle with some serious analysis. There are a lot of (outstanding) exercises and these are critical for a deeper understanding of the material. Happily all of Chapter 7 is devoted to their solutions, and this increases the bookâ€™s already considerable value as a source for self-study.
Also, Dacorogna claims that even if his own focus in Introduction to the Calculus of Variations is on â€œmathematical applications,â€ the book is still accessible to folks from physics, engineering, biology, etc. Letâ€™s just reply â€œYeah, right â€¦â€ to this: itâ€™s hard-core mathematics, make no mistake! But as a more sophisticated introduction to the calculus of variations itâ€™s a very beautiful treatment, and will reward the diligent reader with a solid introduction to a great and grand subject and to a lot of beautiful hard analysis.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.