Introduction To The Calculus of Variations And Its Applications

657 Pages·1995·20.083 MB·\657

by  Frederic Wan (Author)

INTRODUCTION TO
THE CALCULUS OF
VARIATIONS AND
ITS APPLICATIONS

INTRODUCTION TO
THE CALCULUS OF
VARIATIONS AND
ITS APPLICATIONS
Frederick Y.M. Wan
University ot California, Irvine
CRC Press
Taylor &. Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A CHAPMAN & HALL BOOK
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Copyright © 1995
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Library of Congress Cataloging-in-Publication Data
Wan, Frederic Y.M.
Introduction to the calculus of variations and its applications/Frederic Y.M. Wan.
p. cm.
Includes bibliographical references.
ISBN 0-412-05141-9
1. Calculus of variations. I. Title.
QA315.W34 1993
515’.64—dc20 93-30157
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To my teacher, Eric Reissner,
who made it all possible.

Contents
Page
Preface xiii
1. The Basic Problem
1. Introduction 1
2. Some Examples 3
3. The Euler Differential Equation 9
4. Integration of the Euler Differential Equation 14
5. The Brachistochrone Problem 22
6. Piecewise-Smooth Extremals 26
7. Exercises 28
2. Piecewise-Smooth Extremals
1. Piecewise-Smooth Solution for the Basic Problem 34
2. The Euler-Lagrange Equation 37
3. Several Unknowns 39
4. Parametric Form 42
5. Erdmann’s Comer Conditions 45
6. The Ultra-Differentiated Form 48
7. Minimal Surface of Revolution 49
8. Maximum Rocket Height 53
9. Exercises 55
3. Modifications of the Basic Problem
1. The Variational Notation 58
2. Euler Boundary Conditions 61
3. Free Boundary Problems 65
Contents
4. Free and Constrained End Points 68
5. Higher Derivatives 71
6. Other End Conditions 76
7. Exercises 79
A Weak Minimum
1. The Legendre Condition 85
2. Jacobi’s Test 89
3. Conjugate Points 92
4. Sufficiency 96
5. Several Unknowns 99
6. Convex Integrand 104
7. Global Minimum 107
8. Exercises 109
A Strong Minimum
1. A Weak Minimum May Not Be the True Minimum 113
2. The Weierstrass Excess Function 115
3. The Figurative 117
4. Fields of Extremals 120
5. Sufficiency 124
6. An Illustrative Example 128
7. Hilbert’s Integral 130
8. Several Unknowns 133
9. Exercises 135
Appendix 137
The Hamiltonian
1. The Legendre Transformation and Hamiltonian Systems 139
2. Hamilton’s Principle 142
3. Canonical Transformations 145
4. The Hamilton-Jacobi Equation 148
5. Solutions of the Hamilton-Jacobi Equation 152
6. The Method of Additive Separation 155
7. Hamilton’s Principal Function 160
8. Exercises 164
Lagrangian Mechanics
1. Generalized Coordinates 167
2. Coordinate Transformations 170
Contents ix
3. Holonomic Constraints 176
4. Poisson Brackets 180
5. Variationally Invariant Lagrangians 182
6. Noether’s Theorem 184
7. Generators for Variationally Invariant Lagrangians 188
8. Relativistic Mechanics 191
9. Exercises 194
8. Direct Methods
1. The Rayleigh-Ritz Method 198
2. Completeness and Minimizing Sequence 201
3. A Weighted Least-Squares Approximation 205
4. Inhomogeneous End Conditions 207
5. Piecewise Linear Finite Elements 212
6. The Finite Element Method 216
7. Duality 218
8. The Inverse Problem 223
9. Weak Solutions 228
10. Exercises 230
9. Dynamic Programming
1. The Shortest Route Problem 233
2. Backward Recursion 238
3. The Knapsack Problem 242
4. Forward Recursion 245
5. Intermediate Knapsack Capacities 248
6. Vector- and Continuous-State Variables 250
7. The Variational Problem 257
8. Exercises 261
10. Isoperimetric Constraints
1. The Shape of the Hanging Chain 266
2. Normal Isoperimetric Problems and a Duality 270
3. Eigenvalue Problems and Mechanical Vibration 273
4. Variational Formulation of Sturm-Liouville Problems 277
5. The Rayleigh Quotient 280
6. Higher Eigenvalues 283
7. Mixed End Conditions 285
8. Optimal Harvesting of a Uniform Forest 286
9. Exercises 289
Appendix 293