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Elementary Symplectic Topology and Mechanics

Franco Cardin
Publication Date: 
Number of Pages: 
Lecture Notes of the Unione Matematica Italiana
[Reviewed by
William J. Satzer
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This book is mostly about aspects of the Hamilton-Jacobi approach to mechanics. A variety of related topics appear: symplectic geometry and topology, the calculus of variations, bits of algebraic topology and finally Lusternik-Schnirelman and Morse theories.

The word “elementary” of the title is misleading. Whatever the author’s intentions, much of the book is not “didactically simple”. A good deal of it seems to be aimed at either specialists or others who are largely familiar with the basic ideas. A reader with the appropriate background will find quite a number of interesting ideas, connections and insights. But the organization and selection of topics are quirky. Much of it seems to follow a pattern the author has in mind, one that never quite comes through to the reader.

The book begins with a review of differential geometry from the viewpoint of mechanics: tangent and cotangent bundles, differential forms, the Lie derivative and Riemannian geometry. Next is an introduction to symplectic manifolds that, after a quick look at the basics, moves on to generating functions and then lightly touches on symplectic geometry with a look at one version of Gromov’s non-squeezing theorem. Here the author makes some interesting comments on consequences of this theorem for mechanical systems, but they are unfortunately brief and the author does not follow up on them.

Two primary topics occupy much of the rest of the book. The first is what the author calls the Cauchy problem for Hamilton-Jacobi equations. The second (the longest in the book) is a discussion of Lusternik-Schnirelman and Morse theories and their application to variational solutions of the Hamilton-Jacobi equations.

Specialists may be interested in the author’s take on the interplay between Lusternik-Schnirelman theory and Morse theory. These can be thought of as powerful complementary modern approaches to the calculus of variations. The author views them as independent approaches developed in separated mathematical communities that have fruitful unexplored interrelationships.

The signs of the book’s origin as draft notes for a course are only too apparent. The writing is uneven, detailed in one area then sparse in another, and too often there are no transitions between subjects.

The author has provided a strong bibliography. Unfortunately there is no index, and the book badly needs one. There are also no exercises. In several places throughout the book the translation from Italian to English is imperfect and this leads to more than a few very puzzling sentences.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Beginning.- Notes on Differential Geometry.- Symplectic Manifolds.- Poisson brackets environment.- Cauchy Problem for H-J equations.- Calculus of Variations and Conjugate Points.- Asymptotic Theory of Oscillating Integrals.- Lusternik-Schnirelman and Morse.- Finite Exact Reductions.- Other instances.- Bibliography.