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The Calculus of Variations

Publisher: 
Springer Verlag
Number of Pages: 
290
Price: 
69.95
ISBN: 
0-387-40247-0

Somebody once said of opera, "You either love it or you don't understand it." The same could be said of the calculus of variations. Many, if not most mathematicians have never studied the subject, and so could hardly be expected to love it. Bruce van Brunt shows his love of the subject in his new book The Calculus of Variations, part of the Universitext series from Springer.

All accounts of the calculus of variations start from the same foundation, the Euler-Lagrange equation, a differential equation that provides a condition necessary for a curve to be an optimal curve. Van Brunt takes the practical approach and makes the most of this necessary condition, rather than following a more theoretical tack and dwelling on sufficient conditions. In fact, this one emphasis can be used as a kind of litmus test to separate those books aimed at engineers and undergraduates from those written for mathematicians and graduate students. This book is crafted for engineers and undergraduates.

Van Brunt gives us a nice historical introduction to the calculus of variations. The topic has a long, if discontinuous history going to the Bernoulli brothers in the 1690s. He quotes a 1927 textbook that the subject "attracted a rather fickle attention at more or less isolated intervals in its growth." The giants of the 17th and 18th centuries, Newton, Leibniz, Euler, Lagrange, Legendre, gave us the first principles and the classical examples. It grew more slowly in the 19th century, only to be reawakened by Hilbert's 23rd problem. In the 20th century, it enjoyed the attention of Noether, Lebesgue, Hadamard and Carathéodory, among others.

All textbooks start from the same flagship examples, the catenary, the brachistochrone, and Dido's isoperimetric problem, and the same special cases, no explicit dependence on x, or no dependence on y. These are the "best" examples, the ones that arise from interesting problems and are at the same time "doable." They are also scarce resources, and van Brunt uses them wisely. He doesn't overuse them, but instead supports them with a rich assortment of lesser examples. The exercises have obviously been polished and sharpened in the classroom.

The subject itself is not so difficult, but the calculus of variations does have a substantial list of prerequisites, especially differential equations and classical mechanics. It would turn into a graduate level course if we also required normed spaces, but in exchange we would get to learn about the differences between weak and strong extremals and more about sufficient conditions. Van Brunt knows where that line is and stops where we can see what is beyond without crossing the line.

In total, this is a well crafted, reasonably priced book that would be a fine introduction to a fascinating subject that not enough mathematicians know about.


Ed Sandifer is a Professor of Mathematics at Western Connecticut State University in Danbury, Connecticut. His research interests are mostly historical, especially Leonhard Euler, and he has run the Boston Marathon every year since 1973. He can be reached at SandiferE@WCSU.edu

Date Received: 
Wednesday, September 1, 2004
Reviewable: 
Yes
Include In BLL Rating: 
No
Bruce van Brunt
Series: 
Universitext
Publication Date: 
2004
Format: 
Hardcover
Category: 
Textbook
Ed Sandifer
05/1/2004
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Catenary and Brachystochrone Problems . . . . . . . . . . . . . . . 3
1.2.1 The Catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Brachystochrones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Hamilton's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Some Variational Problems from Geometry . . . . . . . . . . . . . . . . . 14
1.4.1 Dido's Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.3 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Optimal Harvest Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 The First Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 The Finite-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 The Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Some Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.1 Case I: No Explicit y Dependence . . . . . . . . . . . . . . . . . . . 36
2.3.2 Case II: No Explicit x Dependence . . . . . . . . . . . . . . . . . . 38
2.4 A Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Invariance of the Euler-Lagrange Equation . . . . . . . . . . . . . . . . . . 44
2.6 Existence of Solutions to the Boundary-Value Problem* . . . . . . 49
3 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Functionals Containing Higher-Order Derivatives . . . . . . . . . . . . 55
3.2 Several Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Two Independent Variables* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 The Inverse Problem* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
XII Contents
4 Isoperimetric Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 The Finite-Dimensional Case and Lagrange Multipliers . . . . . . . 73
4.1.1 Single Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.2 Multiple Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.3 Abnormal Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 The Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Some Generalizations on the Isoperimetric Problem . . . . . . . . . . 94
4.3.1 Problems Containing Higher-Order Derivatives . . . . . . . . 95
4.3.2 Multiple Isoperimetric Constraints . . . . . . . . . . . . . . . . . . . 96
4.3.3 Several Dependent Variables . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Applications to Eigenvalue Problems* . . . . . . . . . . . . . . . . . . . . . 103
5.1 The Sturm-Liouville Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 The First Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3 Higher Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6 Holonomic and Nonholonomic Constraints . . . . . . . . . . . . . . . . . 119
6.1 Holonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Nonholonomic Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.3 Nonholonomic Constraints in Mechanics* . . . . . . . . . . . . . . . . . . . 131
7 Problems with Variable Endpoints . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.1 Natural Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.3 Transversality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 The Hamiltonian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.1 The Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2 Hamilton's Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.3 Symplectic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.4 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4.1 The General Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.4.2 Conservative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.5 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8.5.1 The Method of Additive Separation . . . . . . . . . . . . . . . . . . 185
8.5.2 Conditions for Separable Solutions* . . . . . . . . . . . . . . . . . . 190
9 Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 Variational Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.3 Noether's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4 Finding Variational Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Contents XIII
10 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.1 The Finite-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
10.2 The Second Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10.3 The Legendre Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.4 The Jacobi Necessary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10.4.1 A Reformulation of the Second Variation . . . . . . . . . . . . . 232
10.4.2 The Jacobi Accessory Equation . . . . . . . . . . . . . . . . . . . . . 234
10.4.3 The Jacobi Necessary Condition . . . . . . . . . . . . . . . . . . . . . 237
10.5 A Sufficient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.6 More on Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
10.6.1 Finding Conjugate Points . . . . . . . . . . . . . . . . . . . . . . . . . . 245
10.6.2 A Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . . . . 249
10.6.3 Saddle Points* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
10.7 Convex Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
A Analysis and Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 261
A.1 Taylor's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
A.2 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
A.3 Theory of Ordinary Differential Equations . . . . . . . . . . . . . . . . . . 268
B Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
B.1 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
B.2 Banach and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
Publish Book: 
Modify Date: 
Thursday, August 26, 2010

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