Fundamental Theories and Their Applications of the Calculus of Variations

Foreword
Preface
Introduction
Contents
About the Authors
1 Preliminaries
	1.1 The Taylor Formulae
		1.1.1 Case of a Function of One Variable
		1.1.2 Cases of Functions of Several Variables
	1.2 Integrals with Parameters
	1.3 Fundamentals of the Theory of Field
		1.3.1 Directional Derivative and Gradient
		1.3.2 Flux and Divergence of Vector Field
		1.3.3 The Gauss Theorem and Green’s Formulae
		1.3.4 Circulation and Rotation of Vector Field
		1.3.5 The Stokes Theorem
		1.3.6 The United Gauss Formula Expressed by Gradient, Divergence and Rotation
	1.4 Coordinate Transformations Between Rectangular Coordinate System and Polar Coordinates
	1.5 Fundamental Lemmas of the Calculus of Variations
	1.6 Summation Convention, Kronecker Delta and Permutation Symbols
	1.7 Basic Conceptions of Tensors
		1.7.1 Rotation Transformations of Rectangle Coordinates
		1.7.2 The Cartesian Second Order Tensors
		1.7.3 Algebraic Operations of Cartesian Tensors
		1.7.4 Quotient Laws of Tensors
		1.7.5 Principal Axes, Characteristic Values and Invariants of Second Order Tensors
		1.7.6 Differential Operations of the Cartesian Tensors
	1.8 Some Inequalities in Common Use
	1.9 Introduction to the Famous Scientists
2 Variational Problems with Fixed Boundaries
	2.1 Examples of the Classical Variational Problems
	2.2 Fundamental Conceptions of the Calculus of Variations
	2.3 Variations of the Simplest Functionals and Necessary Conditions of Extrema of Functionals
	2.4 The Euler Equations of the Simplest Functional
	2.5 Several Special Cases of the Euler Equation and Their Integrals
	2.6 Variational Problems Depending on Several Functions of One Variable
	2.7 Variational Problems Depending on Higher Order Derivatives
	2.8 Variational Problems Depending on Functions of Several Variables
	2.9 Variational Problems of Complete Function
	2.10 Invariance of the Euler Equation
	2.11 Introduction to the Famous Scientists
3 Sufficient Conditions of Extrema of Functionals
	3.1 Extremal Curve Fields
	3.2 The Jacobi Conditions and Jacobi Equation
	3.3 The Weierstrass Functions and Weierstrass Conditions
	3.4 The Legendre Conditions
	3.5 Sufficient Conditions of Extrema of Functionals
		3.5.1 The Weierstrass Sufficient Conditions
		3.5.2 The Legendre Sufficient Conditions
	3.6 Higher Order Variations of Functionals
	3.7 Introduction to the Famous Scientists
4 Problems with Variable Boundaries
	4.1 Variational Problems of the Simplest Functional
	4.2 Variational Problems of Functionals with Several Functions
	4.3 Variational Problems of Functionals with Higher Order Derivatives
		4.3.1 Cases of Functionals with One Unknown Function and Its Second Derivative
		4.3.2 Cases of Functionals with One Unknown Function and Its Several Order Derivatives
		4.3.3 Cases of Functionals with Several Unknown Functions and Their Several Order Derivatives
	4.4 Variational Problems of Functionals with Functions of Several Variables
	4.5 Extremal Curves with Cuspidal Points
	4.6 One-Sided Variational Problems
	4.7 Introduction to the Famous Scientists
5 Variational Problems of Conditional Extrema
	5.1 Variational Problems with Holonomic Constraints
	5.2 Variational Problems with Differential Constraints
	5.3 Isoperimetric Problems
	5.4 Extremal Problems of Mixed Type Functionals
		5.4.1 Extremal Problems of Simple Mixed Type Functionals
		5.4.2 Euler Equations of 2-D, 3-D and n-D Problems
	5.5 Introduction to the Famous Scientists
6 Variational Problems in Parametric Forms
	6.1 Parametric Forms of Curves and Homogeneous Condition
	6.2 Isoperimetric Problems in Parametric Forms and Geodesic Line
	6.3 Extrema of Functionals with Variable Boundaries and Parametric Forms
7 Variational Principles
	7.1 Sets and Mappings
	7.2 Sets and Spaces
	7.3 Normal Orthogonal System and Fourier Series
	7.4 Operators and Functionals
	7.5 Derivatives of Functionals
	7.6 Variational Principles of Operator Equations
	7.7 Variational Problems of Equivalence with Boundary Value Problem of Self Conjugate Ordinary Differential Equation
	7.8 Variational Problems of Equivalence with Boundary Value Problem of Self Conjugate Partial Differential Equation
	7.9 The Friedrichs Inequality and Poincaré Inequality
	7.10 To the Famous Scientists
8 Direct Methods of Variational Problems
	8.1 Minimizing (Maximizing) Sequence
	8.2 The Euler Finite Difference Method
	8.3 The Ritz Method
	8.4 The Kantorovich Method
	8.5 The Galerkin Method
	8.6 The Least Square Method
	8.7 Eigenvalues and Eigenfunctions of Operator Equations
	8.8 Introduction to the Famous Scientists
9 Variational Principles in Mechanics and Their Applications
	9.1 Fundamental Conceptions in Mechanics
		9.1.1 System of Mechanics
		9.1.2 Constraints and Their Classification
		9.1.3 Actual Displacement and Virtual Displacement
		9.1.4 Relations Between Strains and Displacements
		9.1.5 Work and Energies
	9.2 Principle of Virtual Displacement
		9.2.1 Principle of Virtual Displacement for System of Particles
		9.2.2 Principle of Generalized Virtual Displacement for Elastic Body
		9.2.3 Principle of Generalized Virtual Displacement for Elastic Body
	9.3 Principle of the Minimum Potential Energy
	9.4 Principle of Complementary Virtual Work
	9.5 Principle of the Minimum Complementary Energy
	9.6 The Hamilton Principles and Their Applications
		9.6.1 The Hamilton Principle of System of Particles
		9.6.2 The Hamilton Principle of Elastic Body
	9.7 The Hamilton’s Canonical Equations
	9.8 The Hellinger-Reissner Generalized Variational Principles
	9.9 The Hu Haichang-Kyuichiro Washizu Generalized Variational Principles
	9.10 The Maupertuis-Lagrange Principle of Least Action
	9.11 Introduction to the Famous Scientists
10 Variational Problems of Functionals with Vector, Tensor and Hamiltonian Operators
	10.1 Basic Properties of the Tensor Inner Product Operations and Fundamental Lemma of the Variation of Functional with Tensors
	10.2 The Euler Equations of Functionals with Vector, Modulus of Vector and Hamiltonian Operators
	10.3 The Euler Equations of Gradient Type Functionals
	10.4 The Euler Equations of Divergence Type Functionals
	10.5 The Euler Equations of Rotation Type Functionals
	10.6 Variational Problems of Functionals with Parallel-Type Inner Product Tensors and Hamiltonian Operators
		10.6.1 Variational Formula Derivations of Gradients, Divergences and Rotations of Parallel-Type Inner Product Tensors
		10.6.2 The Euler Equations and Natural Boundary Conditions of the Functionals with Parallel-Type Inner Product Tensors and Hamiltonian Operators
		10.6.3 Some Examples of the Functionals with Parallel-Type Inner Product Tensors and Hamiltonian Operators
		10.6.4 The Euler Equations of the Functionals with Parallel-Type Inner Product Tensors and the Hamiltonian Operator Trains
		10.6.5 Other Euler Equations of the Functionals with Parallel-Type Inner Product Tensors and the Hamiltonian Operators
	10.7 Variational Problems of Functionals with Series-Type Inner Product Tensors and Hamiltonian Operators
		10.7.1 Variational Formula Derivations of Gradients, Divergences and Rotations of Series-Type Inner Product Tensors
		10.7.2 The Euler Equations and Natural Boundary Conditions of the Functionals with Series-Type Inner Product Tensors and Hamiltonian Operators
		10.7.3 The Euler Equations of the Functionals with Series-Type Inner Product Tensors and the Hamiltonian Operator Trains
		10.7.4 Other Euler Equations of the Functionals with Series-Type Inner Product Tensors and the Hamiltonian Operators
	10.8 Conclusions
	10.9 Introduction to the Famous Scientists
Appendix All Solutions to the Problems
Chapter 1: Solutions to the Problems in Preliminaries
Chapter 2: Solutions to the Problems in Variational Problems with Fixed Boundaries
Chapter 3: Solutions to the Problems in Sufficient Conditions of Extrema of Functionals
Chapter 4: Solutions to the Problems in Problems with Variable Boundaries
Chapter 5: Solutions to the Problems in Variational Problems of Conditional Extrema
Chapter 6: Solutions to the Problems in Variational Problems in Parametric Forms
Chapter 7: Solutions to the Problems in Variational Principles
Chapter 8: Solutions to the Problems in Direct Methods of Variational Problems
Chapter 9: Solutions to the Problems in Variational Principles in Mechanics and Their Applications
Chapter 10: Solutions to the Problems in Variational Problems of Functionals with Vector, Tensor and Hamiltonian Operators
Bibliography
Index