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Variational Analysis: Critical Extremals and Sturmian Extensions

Dover Publications
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Morse theory is a marvelous example of a mathematical subject whose impetus, or initial motivation, is a single set of questions, driving at a single objective. For Morse theory the raison d’être is to have in hand a systematic and computable analytic treatment of singularities, critical points, stationary points, and so on, in various geometric settings. While there are a number of more modern treatments in existence at this point in time, some very fine (see, for example, Nicolaescu's An Invitation to Morse Theory), the benchmark is surely John Milnor’s famous monograph, Morse Theory, dating back to the 1960s: a gem of concision and clarity, rich in modern topological connections. It is still instructive, however, to read the more analytic (or geometrical) writings of the theory’s originator, specifically, Morse’s famous book, The Calculus of Variations in the Large , first published by the AMS in 1934 and in circulation for many subsequent decades: my copy is the fifth edition of this book, dating to 1965, which renders it contemporary to Milnor’s book. It is clear, in any case, that Morse’s original discussion remains relevant.

And, at least to some extent, the work now under review, Morse’s Variational Analysis: Critical Extremals and Sturmian Extensions , fits the same bill. The book first appeared in the 1970s, at a point in time when Morse’s methods had not only firmly established themselves as part of mainstream global analysis, they had already undergone a good deal of evolution: as already indicated, Milnor’s tone-setting algebraic and differential-topological treatment of Morse theory is very different in flavor from Morse’s original formulations, so it is perhaps fair to say that the indicated expositions by Morse and by Milnor represent different ends of a creative and expository spectrum.

Variational Analysis is indeed written in an idiom that hearkens back to Morse’s earlier work of the 1930s, but this is consonant with the fact that Morse’s goal in writing the book refers back to a classical theme first raised in the 19th century, namely, that of extremals. Variational Analysis addresses the question of passing from the minimal extremals treated by Euler, Legendre, Jacobi, and Weierstrass, in their discussion of the calculus of variations, to the more general critical extremals, which are, by definition, “finite extremal arc[s] that [satisfy] the transversality conditions associated with … prescribed boundary conditions.”

Manifestly, then, the focus of Variational Analysis is relatively narrow, certainly as compared with The Calculus of Variations in the Large (no play on words intended). The book closes with a discussion of differential equations of second order and of quadratic forms, which adds a lot to its appeal.

It is appropriate to remember Abel’s dictum : “Read the masters…”

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

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Saturday, September 29, 2007
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Wednesday, December 5, 2007