You are here

A First Course in the Calculus of Variations

Mark Kot
Publisher: 
American Mathematical Society
Publication Date: 
2014
Number of Pages: 
298
Format: 
Paperback
Series: 
Student Mathematical Library 72
Price: 
50.00
ISBN: 
9781470414955
Category: 
Textbook
BLL Rating: 

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

[Reviewed by
William J. Satzer
, on
12/1/2014
]

The calculus of variations has an especially rich history and an almost endless list of interesting problems. This introductory text follows the historical development of the subject and offers the reader a mixture of theory, techniques and applications. The book is aimed at advanced undergraduates and beginning graduate students in mathematics, physics and engineering. The author integrates theory and applications quite deftly with the historical background and gives us a very attractive book.

Johan Bernoulli’s original question in 1696 about a shortest-time-of descent curve marks the unusually clearly defined birth date of the calculus of variations. (The responders to that question and their solutions remind us of the intellectual power that was available at that time: no less than Newton, Leibnitz, l’Hôpital, and Jakob Bernoulli.) The author uses Bernoulli’s question and a few other examples (a tunneling-through-the earth version of the original brachistochrone, then geodesics and minimal surfaces) to introduce the basic ideas of variational problems. The introductory chapter gives a good indication of what’s to come: clear writing, a carefully laid out development, well-chosen line drawings, and a thoughtful selection of recommended reading.

The development begins with the simplest possible situation: a definite integral with one independent and one dependent variable, one derivative, boundary conditions and a desire to maximize or minimize the integral. The author then presents three derivations of the Euler-Lagrange equation (one from Euler, an improvement from Lagrange and a further enhancement from du Bois-Reymond). A necessary condition for a solution is that it satisfy the Euler-Lagrange equation. With this tool in place, a series of case studies help make the basic notions more concrete as the reader sees how solutions emerge in real applications.

The author advances step by step to more advanced topics: higher derivatives, multiple integrals, the second variation, constraints, and problems with variable endpoints. Along the way he begins to introduce a more nuanced consideration of necessary and sufficient conditions for weak relative minima.

This would serve admirably as the text for a course or as a tool for self-study. The exercises are first rate, although few of them are routine. There is an excellent bibliography.

Buy Now


Bill Satzer (wjsatzer@mmm.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.