When Newton, Euler and the Bernoulli brothers invented the calculus of variations around the beginning of the 18th century, Johann Sebastian Bach was writing the world's best music. The title of this book, Six Themes on Variation, is an echo of those times, when Bach's "theme and variation" style of music was so popular.
As the title suggests, Robert Hardt's little book, only 153 pages long, contains six essays by six different authors (none of them Hardt) each touching on the theme of the calculus of variations. Four of the six essays are between 20 and 25 pages long, the outliers being 12 and 48 pages. All are lively introductions to problems that variational methods help solve.
The preface tells us "One November, Rice University hosted a group o thirty undergraduate mathematics majors with the purpose of introducing them to research mathematics and graduate school. The principle part of this introduction was the series of talks and workshops, which all took up some idea or theme from the calculus of variations." This is the volume that resulted when they wrote up their talks and workshops. As such, it is more a volume about mathematics than a volume of mathematics itself. In this, it reminds me of one of my own memorable teachers, J. C. Su, who tried to teach me differential topology. Every month or so, as he was introducing a new topic, he would take a lecture or two to do what he called "propaganda," a beautiful exposition on something like lens spaces or cobordism that would tell us about the interpretations of the theorems we would be proving for the next few weeks, and a kind of advertisement to recruit, or at least tempt graduate students to do further work in the field.
In the same way, Six Themes on Variation is propaganda, in the sense of J. C. Su, designed to tempt undergraduates, first to graduate school, (the editor and four of the six contributors are at Rice) and then to work in the calculus of variations.
My favorite essay was the second, "How Many Equilibria Are There? An Introduction to Morse Theory," by Rice University's Robin Forman. Of all the essays, this was most obviously written as a talk. At least one joke survived the editing process and made it to the final text. Forman shows us how the continuously differentiable functions on a sphere lead to dissections of the sphere into simplices, and conversely. The dissection depends on the structure of equilibrium points of the function. On a sphere, equilibrium can be stable (a minimum), semi-stable (a saddle point) or unstable (a maximum). The number of equilibria of each type is exactly the same as the number of simplices of each dimension in the corresponding dissection.
The same is true of other surfaces, like tori, and of related objects in higher dimensions. Forman doesn't prove any of this, but we see carefully prepared and well-illustrated examples, so we are ready to believe it is true.
Moreover, Forman gives us a variety of references, both elementary and advanced, and including several of Morse's original articles from the 1920's and 1930's, and some by Palais and Smale in the 1960's. We are being urged at least to look at them. Students are not expected to understand them, but to see what groundbreaking mathematics looks like and to aspire to understand it.
I also enjoyed the last essay, "Hold That Light! Modeling of Traffic Flow by Differential Equations," by Barbara Lee Keyfitz of the University of Houston and the Fields Institute. The modeling of traffic has been a notoriously knotty problem in applied mathematics. Techniques that explain, for example, why a traffic jam moves along a highway like a wave, often fail to explain what seem like closely related phenomena.
Keyfitz uses techniques involving ill-posed systems of differential equations, normally not very useful, to describe the traffic congestion that occurs in the idealized environment of an isolated traffic light on a single-lane one-way highway. As I type this review, I look out my window and watch the traffic back up past my house from the traffic light a quarter of a mile away, I wish that someone in my town's Public Works Department would read this chapter. It would have been nice, though, if Keyfitz had taken a little extra time to make more clearly the connection between this mathematics and the calculus of variations.
The longest essay was Michael Wolf's 48-page "Minimal Surfaces, Flat Cone Spheres and Moduli Spaces of Staircases." The climax of this paper was exciting, but overall I thought it was the wrong length. It was too short to provide enough details to understand the topic, but too long to fit in with the character of the rest of the book.
This might have been a good role for Frank Jones' opening essay "Calculus of Variations: What Does 'Variations' Mean?" That was the shortest essay, and the one that comes closest to doing mathematics.
The other two essays, Frank Morgan on bubbles and Steven Cox on vibrating strings, were both entertaining and informative. Both left this reviewer wanting to read a bit more, and both pointed to a plethora of further reading.
Six Themes on Variation is volume 26 of the Student Mathematical Library inaugurated by the American Mathematical Society in 1999. At least ten of those have already been reviewed here at MAA Online. This is a nice little book on many levels. The exposition is entertaining, the interplay between the mathematics and the applications is interesting, and the idea of "advertising" higher mathematics to undergraduates and graduate students seems exciting and productive.
Ed Sandifer (firstname.lastname@example.org
) writes the column How Euler Did It
for MAA Online. He is professor of mathematics at Western Connecticut State University and has run the Boston Marathon 32 times.