Title | Advanced engineering analysis : the calculus of variations and functional analysis with applications in mechanics |
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File Size | 2.3 MB |

Total Pages | 500 |

Contents Preface 1. Basic Calculus of Variations 1.1 Introduction A function in n variables Functionals Minimization of a simple functional using calculus Notation for various types of derivatives 2. Consider the composite function Brief summary of important terms 1.2 Euler’s Equation for the Simplest Problem 1.3 Properties of Extremals of the Simplest Functional 1.4 Ritz’sMethod 1.5 Natural Boundary Conditions 1.6 Extensions to More General Functionals The functional b a f(x, y, y ) dx The functional b f(x, y, y , . . . , y(n)) dx 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 A few technical details Back to the first variation Variational derivative Brief review of important ideas 1.10 Isoperimetric Problems Two problems Quick summary 1.11 General Formof the First Variation 1.12 Movable Ends of Extremals Quick review 1.13 Broken Extremals: Weierstrass–Erdmann Conditions and Related Problems Quick review 1.14 Sufficient Conditions forMinimum Some field theory 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 Elasticity Reissner–Mindlin plate theory Kirchhoff plate theory Interaction of a plate with elastic beams 2.3 Conservation Laws 2.4 Conservation Laws and Noether’s Theorem The simplest case Functional depending on a vector function 2.5 Functionals Depending on Higher Derivatives of y Functional depending on y 2.6 Noether’s Theorem, General Case Functional depending on a function in n variables and its first derivatives Functional depending on vector function in several variables 2.7 Generalizations Divergence invariance Other 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 Matrices as the component representations of tensors and vectors Elements of calculus for vector and tensor fields Fundamental solution of a linear system of ordinary differential equations 3.8 General Terminal Control Problem 3.9 Pontryagin’sMaximum 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 Lp 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 ContractionMapping 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 Rod under tension Free rod Cantilever beam Free beam Membrane with clamped edge Free membrane Elastic body Plate 4.20 Introduction to Spectral Concepts 4.21 The FredholmTheory 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 Rod Beam Plate Elastic body Nonhomogeneous geometrical boundary conditions 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 An energy space for a clamped membrane (dynamic case) Generalized setup The Faedo–Galerkin method Unique solvability of the Cauchy problem for the nth approximation of the Faedo–Galerkin method Convergence of the Faedo–Galerkin method Uniqueness of the generalized solution 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 Classical setup of the problem Generalized setup 5.16 Exercises Appendix A Hints for Selected Exercises Bibliography Index

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