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TitleAdvanced engineering analysis : the calculus of variations and functional analysis with applications in mechanics
File Size2.3 MB
Total Pages500
Table of Contents
                            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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