| Preface |
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1 Basic Calculus of Variations |
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1 | (98) |
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1 | (14) |
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1.2 Euler's Equation for the Simplest Problem |
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15 | (6) |
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1.3 Properties of Extremals of the Simplest Functional |
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21 | (2) |
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23 | (8) |
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1.5 Natural Boundary Conditions |
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31 | (3) |
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1.6 Extensions to More General Functional |
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34 | (9) |
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1.7 Functional Depending on Functions in Many Variables |
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43 | (6) |
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1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order |
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49 | (5) |
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54 | (11) |
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1.10 Isoperimetric Problems |
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65 | (7) |
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1.11 General Form of the First Variation |
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72 | (4) |
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1.12 Movable Ends of Extremals |
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76 | (4) |
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1.13 Broken Extremals: Weierstrass-Erdmann Conditions and Related Problems |
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80 | (5) |
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1.14 Sufficient Conditions for Minimum |
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85 | (9) |
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94 | (5) |
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2 Applications of the Calculus of Variations in Mechanics |
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99 | (60) |
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2.1 Elementary Problems for Elastic Structures |
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99 | (9) |
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2.2 Some Extremal Principles of Mechanics |
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108 | (19) |
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127 | (4) |
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2.4 Conservation Laws and Noether's Theorem |
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131 | (8) |
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2.5 Functionals Depending on Higher Derivatives of y |
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139 | (4) |
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2.6 Noether's Theorem, General Case |
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143 | (4) |
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147 | (6) |
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153 | (6) |
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3 Elements of Optimal Control Theory |
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159 | (56) |
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3.1 A Variational Problem as an Optimal Control Problem |
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159 | (2) |
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3.2 General Problem of Optimal Control |
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161 | (3) |
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3.3 Simplest Problem of Optimal Control |
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164 | (6) |
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3.4 Fundamental Solution of a Linear Ordinary Differential Equation |
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170 | (1) |
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3.5 The Simplest Problem, Continued |
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171 | (2) |
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3.6 Pontryagin's Maximum Principle for the Simplest Problem |
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173 | (4) |
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3.7 Some Mathematical Preliminaries |
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177 | (12) |
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3.8 General Terminal Control Problem |
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189 | (6) |
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3.9 Pontryagin's Maximum Principle for the Terminal Optimal Problem |
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195 | (3) |
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3.10 Generalization of the Terminal Control Problem |
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198 | (4) |
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3.11 Small Variations of Control Function for Terminal Control Problem |
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202 | (3) |
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3.12 A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problem |
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205 | (3) |
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3.13 Optimal Time Control Problems |
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208 | (4) |
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3.14 Final Remarks on Control Problems |
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212 | (2) |
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214 | (1) |
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215 | (144) |
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4.1 A Normed Space as a Metric Space |
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217 | (6) |
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4.2 Dimension of a Linear Space and Separability |
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223 | (4) |
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4.3 Cauchy Sequences and Banach Spaces |
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227 | (11) |
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4.4 The Completion Theorem |
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238 | (4) |
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4.5 Lp Spaces and the Lebesgue Integral |
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242 | (6) |
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248 | (2) |
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250 | (10) |
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4.8 Inner Product Spaces, Hilbert Spaces |
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260 | (4) |
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4.9 Operators and Functionals |
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264 | (5) |
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4.10 Contraction Mapping Principle |
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269 | (7) |
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4.11 Some Approximation Theory |
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276 | (4) |
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4.12 Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem |
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280 | (4) |
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4.13 Basis, Gram-Schmidt Procedure, and Fourier Series in Hilbert Space |
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284 | (7) |
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291 | (7) |
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4.15 Adjoint and Self-Adjoint Operators |
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298 | (6) |
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304 | (7) |
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311 | (4) |
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4.18 On the Sobolev Imbedding Theorem |
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315 | (5) |
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4.19 Some Energy Spaces in Mechanics |
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320 | (17) |
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4.20 Introduction to Spectral Concepts |
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337 | (6) |
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4.21 The Fredholm Theory in Hilbert Spaces |
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343 | (9) |
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352 | (7) |
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5 Applications of Functional Analysis in Mechanics |
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359 | (74) |
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5.1 Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle |
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359 | (5) |
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5.2 Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs |
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364 | (3) |
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5.3 Equilibrium Problem for a Clamped Membrane and its Generalized Solution |
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367 | (2) |
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5.4 Equilibrium of a Free Membrane |
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369 | (2) |
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5.5 Some Other Equilibrium Problems of Linear Mechanics |
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371 | (8) |
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5.6 The Ritz and Bubnov-Galerkin Methods |
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379 | (2) |
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5.7 The Hamilton-Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics |
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381 | (2) |
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5.8 Generalized Setup of Dynamic Problem for Membrane |
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383 | (14) |
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5.9 Other Dynamic Problems of Linear Mechanics |
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397 | (2) |
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399 | (1) |
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5.11 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics |
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400 | (4) |
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5.12 The Spectral Theorem |
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404 | (6) |
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5.13 The Fourier Method, Continued |
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410 | (5) |
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5.14 Equilibrium of a von Karman Plate |
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415 | (10) |
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5.15 A Unilateral Problem |
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425 | (6) |
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431 | (2) |
| Appendix A Hints for Selected Exercises |
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433 | (50) |
| Bibliography |
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483 | (2) |
| Index |
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485 | |
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