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
