The calculus of variations has fallen on hard times in the mathematics curriculum. While it may never have been a standard undergraduate course, many of the larger schools had retained elective courses on variational methods at least into the 1970s. It is something of a shame that these methods — developed by giants (including Euler, Lagrange, Hamilton, Newton, Leibnitz and the Bernoullis) — have fallen from the mathematical curriculum.
Students who study variational methods often do so in physics or engineering courses — in mechanics or optimal control theory, for example. Until recently there have been no modern standalone textbooks published in the area and many current textbooks on advanced engineering mathematics don’t even mention the subject. Yet there is renewed interest driven by applications, especially in optimal control. The calculus of variations is once again a hot topic. The current book is an attractive fresh look at the subject by a professor of mechanical and aerospace engineering.
The author’s goals are clear: he wants to give students a concise introduction to the basic elements of variational methods and then to concentrate on applications. He deliberately chooses to sacrifice depth for breadth because he wants to explore so many applications.
Part 1 provides preliminary material, a concise introduction to the calculus of variations, and a brief survey of approximation methods (Rayleigh-Ritz, Galerkin, and finite element) for solving variational problems when a closed form solution is not possible. The applications are divided into two parts — physical applications and optimization. “Physical applications” include classical mechanics and dynamical systems, optics and electromagnetics, fluid mechanics and some modern physics. “Optimization” incorporates optimal control, image processing and data analysis as well as introduction to numerical grid generation.
A student wishing to get a taste of variational methods could read just two chapters: a very concise introduction to the calculus of variations, and a well-considered discussion of Hamilton’s principle. Anyone wishing to see the scope of applications should browse the second two-thirds of the book.
The focus throughout is heavy on methods and very light on proofs. The author refers the reader to Luenberger’s Optimization by Vector Space Methods for establishing the rigorous basis of variational methods using the tools of functional analysis.
The book is directed at advanced undergraduates or graduate students in science, engineering or applied mathematics. Prerequisites include only knowledge of calculus and ordinary differential equations. Most chapters have a modest number of exercises. There is a very nice bibliography.
Bill Satzer (email@example.com) 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.