Table of Contents
Preface v
1 Basic Calculus of Variations 1
1.1 Introduction 1
1.2 Euler's Equation for the Simplest Problem 15
1.3 Properties of Extremals of the Simplest Functional 21
1.4 Ritz's Method 23
1.5 Natural Boundary Conditions 31
1.6 Extensions to More General Functionals 34
1.7 Functionals Depending on Functions in Many Variables 43
1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order 49
1.9 The First Variation 54
1.10 Isoperimetric Problems 65
1.11 General Form of the First Variation 72
1.12 Movable Ends of Extremals 76
1.13 Broken Extremals: Weierstrass-Erdmann Conditions and Related Problems 80
1.14 Sufficient Conditions for Minimum 85
1.15 Exercises 94
2 Applications of the Calculus of Variations in Mechanics 99
2.1 Elementary Problems for Elastic Structures 99
2.2 Some Extremal Principles of Mechanics 108
2.3 Conservation Laws 127
2.4 Conservation Laws and Noether's Theorem 131
2.5 Functionals Depending on Higher Derivatives of y 139
2.6 Noether's Theorem, General Case 143
2.7 Generalizations 147
2.8 Exercises 153
3 Elements of Optimal Control Theory 159
3.1 A Variational Problem as an Optimal Control Problem 159
3.2 General Problem of Optimal Control 161
3.3 Simplest Problem of Optimal Control 164
3.4 Fundamental Solution of a Linear Ordinary Differential Equation 170
3.5 The Simplest Problem, Continued 171
3.6 Pontryagin's Maximum Principle for the Simplest Problem 173
3.7 Some Mathematical Preliminaries 177
3.8 General Terminal Control Problem 189
3.9 Pontragin's Maximum Principle for the Terminal Optimal Problem 195
3.10 Generalization of the Terminal Control Problem 198
3.11 Small Variations of Control Function for Terminal Control Problem 202
3.12 A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problems 205
3.13 Optimal Time Control Problems 208
3.14 Final Remarks on Control Problems 212
3.15 Exercises 214
4 Functional Analysis 215
4.1 A Normed Space as a Metric Space 217
4.2 Dimension of a Linear Space and Separability 223
4.3 Cauchy Sequences and Banach Spaces 227
4.4 The Completion Theorem 238
4.5 Lp Spaces and the Lebesgue Integral 242
4.6 Sobolev Spaces 248
4.7 Compactness 250
4.8 Inner Product Spaces, Hilbert Spaces 260
4.9 Operators and Functionals 264
4.10 Contraction Mapping Principle 269
4.11 Some Approximation Theory 276
4.12 Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem 280
4.13 Basis, Gram-Schmidt Procedure, and Fourier Series in Hilbert Space 284
4.14 Weak Convergence 291
4.15 Adjoint and Self-Adjoint Operators 298
4.16 Compact Operators 304
4.17 Closed Operators 311
4.18 On the Sobolev Imbedding Theorem 315
4.19 Some Energy Spaces in Mechanics 320
4.20 Introduction to Spectral Concepts 337
4.21 The Fredholm Theory in Hilbert Spaces 343
4.22 Exercises 352
5 Applications of Functional Analysis in Mechanics 359
5.1 Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle 359
5.2 Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs 364
5.3 Equilibrium Problem for a Clamped Membrane and its Generalized Solution 367
5.4 Equilibrium of a Free Membrane 369
5.5 Some Other Equilibrium Problems of Linear Mechanics 371
5.6 The Ritz and Bubnov-Galerkin Methods 379
5.7 The Hamilton-Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics 381
5.8 Generalized Setup of Dynamic Problem for Membrane 383
5.9 Other Dynamic Problems of Linear Mechanics 397
5.10 The Fourier Method 399
5.11 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics 400
5.12 The Spectral Theorem 404
5.13 The Fourier Method, Continued 410
5.14 Equilibrium of a von Kármán Plate 415
5.15 A Unilateral Problem 425
5.16 Exercises 431
Appendix A Hints for Selected Exercises 433
Bibliography 483
Index 485
