| Preface | p. v |
| Basic Calculus of Variations | p. 1 |
| Introduction | p. 1 |
| Euler's Equation for the Simplest Problem | p. 15 |
| Properties of Extremals of the Simplest Functional | p. 21 |
| Ritz's Method | p. 23 |
| Natural Boundary Conditions | p. 31 |
| Extensions to More General Functionals | p. 34 |
| Functionals Depending on Functions in Many Variables | p. 43 |
| A Functional with Integrand Depending on Partial Derivatives of Higher Order | p. 49 |
| The First Variation | p. 54 |
| Isoperimetric Problems | p. 65 |
| General Form of the First Variation | p. 72 |
| Movable Ends of Extremals | p. 76 |
| Broken Extremals: Weierstrass-Erdmann Conditions and Related Problems | p. 80 |
| Sufficient Conditions for Minimum | p. 85 |
| Exercises | p. 94 |
| Applications of the Calculus of Variations in Mechanics | p. 99 |
| Elementary Problems for Elastic Structures | p. 99 |
| Some Extremal Principles of Mechanics | p. 108 |
| Conservation Laws | p. 127 |
| Conservation Laws and Noether's Theorem | p. 131 |
| Functionals Depending on Higher Derivatives of y | p. 139 |
| Noether's Theorem, General Case | p. 143 |
| Generalizations | p. 147 |
| Exercises | p. 153 |
| Elements of Optimal Control Theory | p. 159 |
| A Variational Problem as an Optimal Control Problem | p. 159 |
| General Problem of Optimal Control | p. 161 |
| Simplest Problem of Optimal Control | p. 164 |
| Fundamental Solution of a Linear Ordinary Differential Equation | p. 170 |
| The Simplest Problem, Continued | p. 171 |
| Pontryagin's Maximum Principle for the Simplest Problem | p. 173 |
| Some Mathematical Preliminaries | p. 177 |
| General Terminal Control Problem | p. 189 |
| Pontragin's Maximum Principle for the Terminal Optimal Problem | p. 195 |
| Generalization of the Terminal Control Problem | p. 198 |
| Small Variations of Control Function for Terminal Control Problem | p. 202 |
| A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problems | p. 205 |
| Optimal Time Control Problems | p. 208 |
| Final Remarks on Control Problems | p. 212 |
| Exercises | p. 214 |
| Functional Analysis | p. 215 |
| A Normed Space as a Metric Space | p. 217 |
| Dimension of a Linear Space and Separability | p. 223 |
| Cauchy Sequences and Banach Spaces | p. 227 |
| The Completion Theorem | p. 238 |
| Lp Spaces and the Lebesgue Integral | p. 242 |
| Sobolev Spaces | p. 248 |
| Compactness | p. 250 |
| Inner Product Spaces, Hilbert Spaces | p. 260 |
| Operators and Functionals | p. 264 |
| Contraction Mapping Principle | p. 269 |
| Some Approximation Theory | p. 276 |
| Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem | p. 280 |
| Basis, Gram-Schmidt Procedure, and Fourier Series in Hilbert Space | p. 284 |
| Weak Convergence | p. 291 |
| Adjoint and Self-Adjoint Operators | p. 298 |
| Compact Operators | p. 304 |
| Closed Operators | p. 311 |
| On the Sobolev Imbedding Theorem | p. 315 |
| Some Energy Spaces in Mechanics | p. 320 |
| Introduction to Spectral Concepts | p. 337 |
| The Fredholm Theory in Hilbert Spaces | p. 343 |
| Exercises | p. 352 |
| Applications of Functional Analysis in Mechanics | p. 359 |
| Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle | p. 359 |
| Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs | p. 364 |
| Equilibrium Problem for a Clamped Membrane and its Generalized Solution | p. 367 |
| Equilibrium of a Free Membrane | p. 369 |
| Some Other Equilibrium Problems of Linear Mechanics | p. 371 |
| The Ritz and Bubnov-Galerkin Methods | p. 379 |
| The Hamilton-Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics | p. 381 |
| Generalized Setup of Dynamic Problem for Membrane | p. 383 |
| Other Dynamic Problems of Linear Mechanics | p. 397 |
| The Fourier Method | p. 399 |
| An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics | p. 400 |
| The Spectral Theorem | p. 404 |
| The Fourier Method, Continued | p. 410 |
| Equilibrium of a von Kármán Plate | p. 415 |
| A Unilateral Problem | p. 425 |
| Exercises | p. 431 |
| Hints for Selected Exercises | p. 433 |
| Bibliography | p. 483 |
| Index | p. 485 |
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