第4卷主要论述非线性泛函分析在数学物理中（包括力学、弹性学、塑性学、流体运动学、热力学、统计力学、狭义相对论和广义相对论、宇宙学等）的应用。给出有关的物理背景及有关的基本方程，用泛函分析的经典和现代结果对在物理学发展中起重要作用的重要问题进行深入讨论。是一本沟通物理学和数学的好书。

Preface

Translator's Preface

INTRODUCTION

Mathematics and Physics

APPLICATIONS IN MECHANICS

CHAPTER 58

 Basic Equations of Point Mechanics

 58.1.Notations

 58.2.Lever Principle and Stability of the Scales

 58.3.Perspectives

 58.4.Kepler's Laws and a Look at the History of Astronomy

 58.5.Newton's Basic Equations

 58.6.Changes of the System of Reference and the Role of Inertial Systems

 58.7.General Point System and Its Conserved Quantities

 58.8.Newton's Law of Gravitation and Coulomb's Law of Electrostatics

 58.9.Application to the Motion of Planets

 58.10.Gauss' Principle of Least Constraint and the General Basic

 Equations of Point Mechanics with Side Conditions

 58.11.Principle of Virtual Power

 58.12.Equilibrium States and a General Stability Principle

 58.13.Basic Equations of the Rigid Body and the Main Theorem about the

 Motion of the Rigid Body and Its Equilibrium

 58.14.Foundation of the Basic Equations of the Rigid Body

 58.15.Physical Models, the Expansion of the Universe, and Its Evolution

 after the Big Bang

 58.16.Legendre Transformation and Conjugate Functionals

 58.17.Lagrange Multipliers

 58.18.Principle of Stationary Action

 58.19.Trick of Position Coordinates and Lagrangian Mechanics

 58.20.Hamiltonian Mechanics

 58.21.Poissonian Mechanics and Heisenberg's Matrix Mechanics in

 Quantum Theory

 58.22.Propagation of Action

 58.23.Hamilton-Jacobi Equation

 58.24.Canonical Transformations and the Solution of the Canonical

 Equations via the Hamilton-Jacobi Equation

 58.25.Lagrange Brackets and the Solution of the Hamilton-Jacobi

 Equation via the Canonical Equations

 58.26.Initial-Value Problem for the Hamiiton-Jacobi Equation

 58.27.Dimension Analysis

CHAPTER 59

 Dualism Between Wave and Particle, Preview of Quantum Theory,

 and Elementary Particles

 59.1.Plane Waves

 59.2.Polarization

 59.3.Dispersion Relations

 59.4.Spherical Waves

 59.5.Damped Oscillations and the Frequency-Time Uncertainty Relation

 59.6.Decay of Particles

 59.7.Cross Sections for Elementary Particle Processes and the Main

 Objectives in Quantum Field Theory

 59.8.Dualism Between Wave and Particle for Light

 59.9.Wave Packets and Group Velocity

 59.10.Formulation of a Pal:ticle Theory for a Classical Wave Theory

 59.11.Motivation of the Schrfdinger Equation and Physical Intuition

 59.12.Fundamental Probability Interpretation of Quantum Mechanics

 59.13.Meaning of Eigenfunctions in Quantum Mechanics

 59.14.Meaning of Nonnormalized States

 59.15.Special Functions in Quantum Mechanics

 59.16.Spectrum of the Hydrogen Atom

 59.17.Functional Analytic Treatment of the Hydrogen Atom

 59.18.Harmonic Oscillator in Quantum Mechanics

 59.19.Heisenberg's Uncertainty Relation

 59.20.Pauli Principle, Spin, and Statistics

 59.21.Quantization of the Phase Space and Statistics

 59.22.Pauli Principle and the Periodic System of the Elements

 59.23.Classical Limiting Case of Quantum Mechanics and the

 WKB Method to Compute Quasi-Classical Approximations

 59.24.Energy-Time Uncertainty Relation and Elementary Particles

 59.25.The Four Fundamental Interactions

 59.26.Strength of the Interactions

 APPLICATIONS IN ELASTICITY THEORY

CHAPTER 60

 Elastoplastic Wire

 60.1.Experimental Result

 60.2.Viscoplastic Constitutive Laws

 60.3.Elasto-Viscoplastic Wire with Linear Hardening Law

 60.4.Quasi-Statical Plasticity

 60.5.Some Historical Remarks on Plasticity

CHAPTER 61

 Basic Equations of Nonlinear Elasticity Theory

 61.1.Notations

 61.2.Strain Tensor and the Geometry of Deformations

 61.3.Basic Equations

 61.4.Physical Motivation of the Basic Equations

 61.5.Reduced Stress Tensor and the Principle of Virtual Power

 61.6.A General Variational Principle (Hyperelasticity)

 61.7.Elastic Energy of the Cuboid and Constitutive Laws

 61.8.Theory oflnvariants and the General Structure of Constitutive Laws

 and Stored Energy Functions

 61.9.Existence and Uniqueness in Linear Elastostatics (Generalized

 Solutions)

 61.10.Existence and Uniqueness in Linear Elastodynamics (Generalized

 Solutions)

 61.11.Strongly Elliptic Systems

 61.12.Local Existence and Uniqueness Theorem in Nonlinear Elasticity via

 the Implicit Function Theorem

 61.13.Existence and Uniqueness Theorem in Linear Elastostatics (Classical

 Solutions)

 61.14.Stability and Bifurcation in Nonlinear Elasticity

 61.15.The Continuation Method in Nonlinear Elasticity and an

 Approximation Method

 61.16.Convergence of the Approximation Method

CHAPTER 62

 Monotone Potential Operators and a Class of Models with Nonlinear

 Hooke's Law, Duality and Plasticity, and Polyconvexity

 62.1.Basic Ideas

 62.2.Notations

 62.3.Principle of Minimal Potential Energy, Existence, and Uniqueness

 62.4.Principle of Maximal Dual Energy and Duality

 62.5.Proofs of the Main Theorems

 62.6.Approximation Methods

 62.7.Applications to Linear Elasticity Theory

 62.8.Application to Nonlinear Hencky Material

 62.9.The Constitutive Law for Quasi-Statical Plastic Material

 62.10.Principle of Maximal Dual Energy and the Existence Theorem for

 Linear Quasi-Statical Plasticity

 62.11.Duality and the Existence Theorem for Linear Statical Plasticity

 62.12.Compensated Compactness

 62.13.Existence Theorem for Polyconvex Material

 62.14.Application to Rubberlike Material

 62.15.Proof of Korn's Inequality

 62.16.Legendre Transformation and the Strategy of the General Friedrichs

 Duality in the Calculus of Variations

 62.17.Application to the Dirichlet Problem (Trefftz Duality)

 62.18.Application to Elasticity

CHAPTER 63

 Variational Inequalities and the Signorini Problem for Nonlinear

 Material

 63.1.Existence and Uniqueness Theorem

 63.2.Physical Motivation

CHAPTER 64

 Bifurcation for Variational Inequalities

 64.1.Basic Ideas

 64.2.Quadratic Variational Inequalities

 64.3.Lagrange Multiplier Rule for Variational Inequalities

 64.4.Main Theorem

 64.5.Proof of the Main Theorem

 64.6.Applications to the Bending of Rods and Beams

 64.7.Physical Motivation for the Nonlinear Rod Equation

 64.8.Explicit Solution of the Rod Equation

CHAPTER 65

 Pseudomonotone Operators, Bifurcation, and the von Ktrmhn Plate

 Equations

 65.1.Basic Ideas

 65.2.Notations

 65.3.The von Karmam Plate Equations

 65.4.The Operator Equation

 65.5.Existence Theorem

 65.6.Bifurcation

 65.7.Physical Motivation of the Plate Equations

 65.8.Principle of Stationary Potential Energy and Plates with Obstacles

CHAPTER 66

 Convex Analysis, Maximal Monotone Operators, and Elasto-

 Viscoplastic Material with Linear Hardening and Hysteresis

 66.1.Abstract Model for Slow Deformation Processes

 66.2.Physical Interpretation of the Abstract Model

 66.3.Existence and Uniqueness Theorem

 66.4.Applications

 ……

CHAPTER 67

CHAPTER 68

CHAPTER 69

CHAPTER 70

CHAPTER 71

CHAPTER 72

CHAPTER 73

CHAPTER 74

CHAPTER 75

CHAPTER 76

CHAPTER 77

CHAPTER 78

CHAPTER 79

Index