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Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction

Mark Levi
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 69
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

Mathematics and classical mechanics have had something of an off-and-on relationship over the last century of so. At one point mechanics was a standard part of the mathematics curriculum, but for some time it has been no more than an elective. For the most part, mathematicians have been happy to consign mechanics to physicists. Every now and again it will stage a mathematical comeback, most recently in a relatively abstract form involving symplectic and Poisson structures, moment maps and the machinery of Lie groups and Lie algebras. The tide may be turning again, marked perhaps by Spivak’s recent publication of Physics for Mathematicians, Mechanics I. (Spivak writes of this book and his motivation for writing it, “… I could easily understand symplectic structures, it’s elementary mechanics that I don’t understand.”)

The book under review offers a fresh look at classical mechanics addressed to undergraduates and beginning graduate students. In the author’s view, a book like this should aim to say something original, interesting, and correct, but at least two of the three (preferably, interesting and correct). Originality is particularly challenging in a subject with so much history, but the reader will find at least a few pedagogically original ideas here.

Two basic principles underlie the author’s pedagogy. The first is that ideas come before formulas. The second is that analogy is a powerful but under-utilized tool. While the treatment here is rigorous, the author works hard to engage the reader’s intuition. Going at least as far back as Newton (and perhaps even Archimedes), calculus, geometry and mechanics have been deeply intertwined. The current work re-emphasizes how much geometry and calculus contribute to mechanics and how much mechanics returns the favor.

There are four parts to the book: dynamics, variational principles, optimal control, and foundations of Hamiltonian mechanics. In just about every section the author takes on items that seem mysterious or appear to be pulled out of thin air in other texts and offers clear, direct and intuitive explanations. He manages to present quite a bit in relatively short order, and at a reasonably basic level. There is very little here that not accessible at the undergraduate level. The author expects only a background in elementary physics, vector calculus and basic linear algebra.

One of the most valuable aspects of the book — unfortunately rare among textbooks — is that we see an author in command of his subject who shares not just the bare facts but how he thinks about them and how all the pieces fit together. This is evident, for example, when the author provides a simple heuristic argument and a drawing showing why Hamiltonian mechanics arises from the action functional and its critical points. Similarly, he demystifies the ability of objects (and light) to take the path of least action by drawing attention to the basic properties of phase cancellation.

Each chapter has a set of exercises, most of them moderate in difficulty. Answers or hints are sometimes provided.

This is as good a text in introductory mechanics as I have seen. It is not as comprehensive as some, but it has the added benefit of incorporating related material from the calculus of variations and optimal control. It is also far better than most at engaging the student’s intuition. 

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.