非线性泛函分析及其应用：在数学物理中的应用（第4卷）
目 录内容简介
Preface
Translators 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
查看完整
Translators 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
查看完整
目 录内容简介
《非线性泛函分析及其应用，第1卷，不动点定理》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。
目 录内容简介
Preface
Translators 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. Keplers Laws and a Look at the History of Astronomy
58.5. Newtons 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. Newtons Law of Gravitation and Coulombs 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 Heisenbergs 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. Heisenbergs 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
Hookes 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 Korns 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
^ 收 起
Translators 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. Keplers Laws and a Look at the History of Astronomy
58.5. Newtons 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. Newtons Law of Gravitation and Coulombs 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 Heisenbergs 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. Heisenbergs 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
Hookes 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 Korns 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
^ 收 起
目 录内容简介
《非线性泛函分析及其应用，第1卷，不动点定理》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。
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