# The Calculus of Variations

• Bruce van Brunt
Textbook

Part of the Universitext book series (UTX)

1. Front Matter
Pages I-XIII
2. Pages 1-22
3. Pages 23-54
4. Pages 55-72
5. Pages 73-101
6. Pages 103-118
7. Pages 119-133
8. Pages 135-157
9. Pages 159-200
10. Pages 201-220
11. Pages 221-260
12. Back Matter
Pages 261-290

### Introduction

The calculus of variations has a long history of interaction with other branches of mathematics such as geometry and differential equations, and with physics, particularly mechanics. More recently, the calculus of variations has found applications in other fields such as economics and electrical engineering. Much of the mathematics underlying control theory, for instance, can be regarded as part of the calculus of variations.

This book is an introduction to the calculus of variations for mathematicians and scientists. The reader interested primarily in mathematics will find results of interest in geometry and differential equations. I have paused at times to develop the proofs of some of these results, and discuss briefly various topics not normally found in an introductory book on this subject such as the existence and uniqueness of solutions to boundary-value problems, the inverse problem, and Morse theory. I have made “passive use” of functional analysis (in particular normed vector spaces) to place certain results in context and reassure the mathematician that a suitable framework is available for a more rigorous study. For the reader interested mainly in techniques and applications of the calculus of variations, I leavened the book with numerous examples mostly from physics. In addition, topics such as Hamilton’s Principle, eigenvalue approximations, conservation laws, and nonholonomic constraints in mechanics are discussed. More importantly, the book is written on two levels. The technical details for many of the results can be skipped on the initial reading. The student can thus learn the main results in each chapter and return as needed to the proofs for a deeper understanding.

### Keywords

Applied Mathematics Calculus of Variations Eigenvalue calculus differential equation equation function geometry linear optimization mathematics theorem variable

#### Authors and affiliations

• Bruce van Brunt
• 1
1. 1.Institute of Fundamental Sciences Palmerston North CampusMassey UniversityPalmerston NorthNew Zealand

### Bibliographic information

• DOI https://doi.org/10.1007/b97436
• Copyright Information Springer-Verlag New York, Inc. 2004
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-40247-5
• Online ISBN 978-0-387-21697-3
• Buy this book on publisher's site