Advanced Engineering Analysis: The Calculus Of Variations And Functional Analysis With Applications In Mechanics The Calculus of Variations and Functional Analysis with Applications in Mechanics

__Advanced Engineering Analysis__ is a textbook on modern engineering analysis, covering the calculus of variations, functional analysis, and control theory, as well as applications of these disciplines to mechanics. The book offers a brief and concise, yet complete explanation of essential theory and applications. It contains exercises with hints and solutions, ideal for self-study. Readership: Academic and industry: engineers, students; advanced undergraduate in the field of mechanical engineering Contents 8 Preface 6 1. Basic Calculus of Variations 12 1.1 Introduction 12 A function in n variables 15 Functionals 17 Minimization of a simple functional using calculus 21 Notation for various types of derivatives 24 2. Consider the composite function 25 Brief summary of important terms 26 1.2 Euler’s Equation for the Simplest Problem 26 1.3 Properties of Extremals of the Simplest Functional 32 1.4 Ritz’sMethod 34 1.5 Natural Boundary Conditions 42 1.6 Extensions to More General Functionals 45 The functional b a f(x, y, y ) dx 45 The functional b f(x, y, y , . . . , y(n)) dx 47 1.7 Functionals Depending on Functions in Many Variables 54 1.8 A Functional with Integrand Depending on Partial Derivatives of Higher Order 60 1.9 The First Variation 65 A few technical details 66 Back to the first variation 68 Variational derivative 72 Brief review of important ideas 75 1.10 Isoperimetric Problems 76 Two problems 79 Quick summary 82 1.11 General Formof the First Variation 83 1.12 Movable Ends of Extremals 87 Quick review 90 1.13 Broken Extremals: Weierstrass–Erdmann Conditions and Related Problems 91 Quick review 96 1.14 Sufficient Conditions forMinimum 96 Some field theory 102 1.15 Exercises 105 2. Applications of the Calculus of Variations in Mechanics 110 2.1 Elementary Problems for Elastic Structures 110 2.2 Some Extremal Principles of Mechanics 119 Elasticity 119 Reissner–Mindlin plate theory 126 Kirchhoff plate theory 130 Interaction of a plate with elastic beams 131 2.3 Conservation Laws 138 2.4 Conservation Laws and Noether’s Theorem 142 The simplest case 142 Functional depending on a vector function 147 2.5 Functionals Depending on Higher Derivatives of y 150 Functional depending on y 150 2.6 Noether’s Theorem, General Case 154 Functional depending on a function in n variables and its first derivatives 154 Functional depending on vector function in several variables 157 2.7 Generalizations 158 Divergence invariance 158 Other generalizations 162 2.8 Exercises 164 3. Elements of Optimal Control Theory 170 3.1 A Variational Problem as an Optimal Control Problem 170 3.2 General Problem of Optimal Control 172 3.3 Simplest Problem of Optimal Control 175 3.4 Fundamental Solution of a Linear Ordinary Differential Equation 181 3.5 The Simplest Problem, Continued 182 3.6 Pontryagin’s Maximum Principle for the Simplest Problem 184 3.7 Some Mathematical Preliminaries 188 Matrices as the component representations of tensors and vectors 188 Elements of calculus for vector and tensor fields 195 Fundamental solution of a linear system of ordinary differential equations 198 3.8 General Terminal Control Problem 200 3.9 Pontryagin’sMaximum Principle for the Terminal Optimal Problem 206 3.10 Generalization of the Terminal Control Problem 209 3.11 Small Variations of Control Function for Terminal Control Problem 213 3.12 A Discrete Version of Small Variations of Control Function for Generalized Terminal Control Problem 216 3.13 Optimal Time Control Problems 219 3.14 Final Remarks on Control Problems 223 3.15 Exercises 225 4. Functional Analysis 226 4.1 A Normed Space as a Metric Space 228 4.2 Dimension of a Linear Space and Separability 234 4.3 Cauchy Sequences and Banach Spaces 238 4.4 The Completion Theorem 249 4.5 Lp Spaces and the Lebesgue Integral 253 4.6 Sobolev Spaces 259 4.7 Compactness 261 4.8 Inner Product Spaces, Hilbert Spaces 271 4.9 Operators and Functionals 275 4.10 ContractionMapping Principle 280 4.11 Some Approximation Theory 287 4.12 Orthogonal Decomposition of a Hilbert Space and the Riesz Representation Theorem 291 4.13 Basis, Gram–Schmidt Procedure, and Fourier Series in Hilbert Space 295 4.14 Weak Convergence 302 4.15 Adjoint and Self-Adjoint Operators 309 4.16 Compact Operators 315 4.17 Closed Operators 322 4.18 On the Sobolev Imbedding Theorem 326 4.19 Some Energy Spaces in Mechanics 331 Rod under tension 331 Free rod 333 Cantilever beam 335 Free beam 337 Membrane with clamped edge 338 Free membrane 342 Elastic body 343 Plate 346 4.20 Introduction to Spectral Concepts 348 4.21 The FredholmTheory in Hilbert Spaces 354 4.22 Exercises 363 5. Applications of Functional Analysis in Mechanics 370 5.1 Some Mechanics Problems from the Standpoint of the Calculus of Variations; the Virtual Work Principle 370 5.2 Generalized Solution of the Equilibrium Problem for a Clamped Rod with Springs 375 5.3 Equilibrium Problem for a Clamped Membrane and its Generalized Solution 378 5.4 Equilibrium of a Free Membrane 380 5.5 Some Other Equilibrium Problems of Linear Mechanics 382 Rod 382 Beam 383 Plate 384 Elastic body 386 Nonhomogeneous geometrical boundary conditions 387 5.6 The Ritz and Bubnov–Galerkin Methods 390 5.7 The Hamilton–Ostrogradski Principle and Generalized Setup of Dynamical Problems in Classical Mechanics 392 5.8 Generalized Setup of Dynamic Problem for Membrane 394 An energy space for a clamped membrane (dynamic case) 396 Generalized setup 399 The Faedo–Galerkin method 400 Unique solvability of the Cauchy problem for the nth approximation of the Faedo–Galerkin method 402 Convergence of the Faedo–Galerkin method 405 Uniqueness of the generalized solution 407 5.9 Other Dynamic Problems of Linear Mechanics 408 5.10 The Fourier Method 410 5.11 An Eigenfrequency Boundary Value Problem Arising in Linear Mechanics 411 5.12 The Spectral Theorem 415 5.13 The Fourier Method, Continued 421 5.14 Equilibrium of a von Karman Plate 426 5.15 A Unilateral Problem 436 Classical setup of the problem 436 Generalized setup 438 5.16 Exercises 442 Appendix A Hints for Selected Exercises 444 Bibliography 494 Index 496 Content: 1. Basic calculus of variations. 1.1. Introduction. 1.2. Euler's equation for the simplest problem. 1.3. Properties of extremals of the simplest functional. 1.4. Ritz's method. 1.5. Natural boundary conditions. 1.6. Extensions to more general functionals. 1.7. Functionals depending on functions in many variables. 1.8. A functional with integrand depending on partial derivatives of higher order. 1.9. The first variation. 1.10. Isoperimetric problems. 1.11. General form of the first variation. 1.12. Movable ends of extremals. 1.13. Broken extremals: Weierstrass-Erdmann conditions and related problems. 1.14. Sufficient conditions for minimum. 1.15. Exercises -- 2. Applications of the calculus of variations in mechanics. 2.1. Elementary problems for elastic structures. 2.2. Some extremal principles of mechanics. 2.3. Conservation laws. 2.4. Conservation laws and Noether's theorem. 2.5. Functionals depending on higher derivatives of y. 2.6. Noether's theorem, general case. 2.7. Generalizations. 2.8. Exercises -- 3. Elements of optimal control theory. 3.1. A variational problem as an optimal control problem. 3.2. General problem of optimal control. 3.3. Simplest problem of optimal control. 3.4. Fundamental solution of a linear ordinary differential equation. 3.5. The simplest problem, continued. 3.6. Pontryagin's maximum principle for the simplest problem. 3.7. Some mathematical preliminaries. 3.8. General terminal control problem. 3.9. Pontryagin's maximum principle for the terminal optimal problem. 3.10. Generalization of the terminal control problem. 3.11. Small variations of control function for terminal control problem. 3.12. A discrete version of small variations of control function for generalized terminal control problem. 3.13. Optimal time control problems. 3.14. Final remarks on control problems. 3.15. Exercises. 4. Functional analysis. 4.1. A normed space as a metric space. 4.2. Dimension of a linear space and separability. 4.3. Cauchy sequences and Banach spaces. 4.4. The completion theorem. 4.5. L[symbol] spaces and the Lebesgue integral. 4.6. Sobolev spaces. 4.7. Compactness. 4.8. Inner product spaces, Hilbert spaces. 4.9. Operators and functionals. 4.10. Contraction mapping principle. 4.11. Some approximation theory. 4.12 Orthogonal decomposition of a Hilbert space and the Riesz representation theorem. 4.13. Basis, Gram-Schmidt procedure, and Fourier series in Hilbert space. 4.14. Weak convergence. 4.15. Adjoint and self-adjoint operators. 4.16. Compact operators. 4.17. Closed operators. 4.18. On the Sobolev imbedding theorem. 4.19. Some energy spaces in mechanics. 4.20. Introduction to spectral concepts. 4.21. The Fredholm theory in Hilbert spaces. 4.22. Exercises -- 5. Applications of functional analysis in mechanics. 5.1. Some mechanics problems from the standpoint of the calculus of variations the virtual work principle. 5.2. Generalized solution of the equilibrium problem for a clamped rod with springs. 5.3. Equilibrium problem for a clamped membrane and its generalized solution. 5.4. Equilibrium of a free membrane. 5.5. Some other equilibrium problems of linear mechanics. 5.6. The Ritz and Bubnov-Galerkin methods. 5.7. The Hamilton-Ostrogradski principle and generalized setup of dynamical problems in classical mechanics. 5.8. Generalized setup of dynamic problem for membrane. 5.9. Other dynamic problems of linear mechanics. 5.10. The Fourier method. 5.11. An eigenfrequency boundary value problem arising in linear mechanics. 5.12. The spectral theorem. 5.13. The Fourier method, continued. 5.14. Equilibrium of a von Karman plate. 5.15. A unilateral problem. 5.16. Exercises. Focuses on modern engineering analysis, covering the calculus of variations, functional analysis, and control theory, as well as applications of these disciplines to mechanics. This book offers an explanation of essential theory and applications.