非线性泛函分析及其应用 第4卷 在数学物理中的应用 英文版PDF|Epub|txt|kindle电子书版本网盘下载

非线性泛函分析及其应用 第4卷 在数学物理中的应用 英文版PDF格式电子书版下载

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I NTRODUCTION Mathematics and Physics1

APPLICATIONS IN MECHANICS7

CHAPTER 58 Basic Equations of Point Mechanics9

58.1．Notations10

58.2．Lever Principle and Stability of the Scales14

58.3．Perspectives17

58.4．Kepler's Laws and a Look at the History of Astronomy22

58.5．Newton's Basic Equations25

58.6．Changes of the System of Reference and the Role of Inertial Systems28

58.7．General Point System and Its Conserved Quantities32

58.8．Newton's Law of Gravitation and Coulomb's Law of Electrostatics35

58.9．Application to the Motion of Planets38

58.10．Gauss'Principle of Least Constraint and the General Basic Equations of Point Mechanics with Side Conditions45

58.11．Principle of Virtual Power48

58.12．Equilibrium States and a General Stability Principle50

58.13．Basic Equations of the Rigid Body and the Main Theorem about the Motion of the Rigid Body and Its Equilibrium52

58.14．Foundation of the Basic Equations of the Rigid Body55

58.15．Physical Models,the Expansion of the Universe,and Its Evolution after the Big Bang57

58.16．Legendre Transformation and Conjugate Functionals65

58.17．Lagrange Multipliers67

58.18．Principle of Stationary Action69

58.19．Trick of Position Coordinates and Lagrangian Mechanics70

58.20．Hamiltonian Mechanics72

58.21．Poissonian Mechanics and Heisenberg's Matrix Mechanics in Quantum Theory77

58.22．Propagation of Action81

58.23．Hamilton-Jacobi Equation82

58.24．Canonical Transformations and the Solution of the Canonical Equations via the Hamilton-Jacobi Equation83

58.25．Lagrange Brackets and the Solution of the Hamilton-Jacobi Equation via the Canonical Equations84

58.26．Initial-Value Problem for the Hamilton Jacobi Equation87

58.27．Dimension Analysis89

CHAPTER 59 Dualism Between Wave and Particle,Preview of Quantum Theory,and Elementary Particles98

59.1．Plane Waves99

59.2．Polarization101

59.3．Dispersion Relations102

59.4．Spherical Waves103

59.5．Damped Oscillations and the Frequency-Time Uncertainty Relation104

59.6．Decay of Particles105

59.7．Cross Sections for Elementary Particle Processes and the Main Objectives in Quantum Field Theory106

59.8．Dualism Between Wave and Particle for Light107

59.9．Wave Packets and Group Velocity110

59.10．Formulation of a Particle Theory for a Classical Wave Theory111

59.11．Motivation of the Schr？dinger Equation and Physical Intuition112

59.12．Fundamental Probability Interpretation of Quantum Mechanics113

59.13．Meaning of Eigenfunctions in Quantum Mechanics114

59.14．Meaning of Nonnormalized States116

59.15．Special Functions in Quantum Mechanics117

59.16．Spectrum of the Hydrogen Atom118

59.17．Functional Analytic Treatment of the Hydrogen Atom121

59.18．Harmonic Oscillator in Quantum Mechanics122

59.19．Heisenberg's Uncertainty Relation123

59.20．Pauli Principle,Spin,and Statistics125

59.21．Quantization of the Phase Space and Statistics126

59.22．Pauli Principle and the Periodic System of the Elements127

59.23．Classical Limiting Case of Quantum Mechanics and the WKB Method to Compute Quasi-Classical Approximations129

59.24．Energy-Time Uncertainty Relation and Elementary Particles130

59.25．The Four Fundamental Interactions134

59.26．Strength of the Interactions136

APPLICATIONS IN ELASTICITY THEORY143

CHAPTER 60 Elastoplastic Wire145

60.1．Experimental Result147

60.2．Viscoplastic Constitutive Laws149

60.3．Elasto-Viscoplastic Wire with Linear Hardening Law151

60.4．Quasi-Statical Plasticity154

60.5．Some Historical Remarks on Plasticity155

CHAPTER 61 Basic Equations of Nonlinear Elasticity Theory158

61.1．Notations166

61.2．Strain Tensor and the Geometry of Deformations168

61.3．Basic Equations176

61.4．Physical Motivation of the Basic Equations180

61.5．Reduced Stress Tensor and the Principle of Virtual Power184

61.6．A General Variational Principle (Hyperelasticity)190

61.7．Elastic Energy of the Cuboid and Constitutive Laws198

61.8．Theory of Invariants and the General Structure of Constitutive Laws and Stored Energy Functions202

61.9．Existence and Uniqueness in Linear Elastostatics (Generalized Solutions)209

61.10．Existence and Uniqueness in Linear Elastodynamics (Generalized Solutions)212

61.11．Strongly Elliptic Systems213

61.12．Local Existence and Uniqueness Theorem in Nonlinear Elasticity via the Implicit Function Theorem215

61.13．Existence and Uniqueness Theorem in Linear Elastostatics (Classical Solutions)221

61.14．Stability and Bifurcation in Nonlinear Elasticity221

61.15．The Continuation Method in Nonlinear Elasticity and an Approximation Method224

61.16．Convergence of the Approximation Method227

CHAPTER 62 Monotone Potential Operators and a Class of Models with Nonlinear Hooke's Law,Duality and Plasticity,and Polyconvexity233

62.1．Basic Ideas234

62.2．Notations242

62.3．Principle of Minimal Potential Energy,Existence,and Uniqueness244

62.4．Principle of Maximal Dual Energy and Duality245

62.5．Proofs of the Main Theorems247

62.6．Approximation Methods252

62.7．Applications to Linear Elasticity Theory255

62.8．Application to Nonlinear Hencky Material256

62.9．The Constitutive Law for Quasi-Statical Plastic Material257

62.10．Principle of Maximal Dual Energy and the Existence Theorem for Linear Quasi-Statical Plasticity259

62.11．Duality and the Existence Theorem for Linear Statical Plasticity262

62.12．Compensated Compactness264

62.13．Existence Theorem for Polyconvex Material273

62.14．Application to Rubberlike Material277

62.15．Proof of Korn's Inequality278

62.16．Legendre Transformation and the Strategy of the General Friedrichs Duality in the Calculus of Variations284

62.17．Application to the Dirichlet Problem (Trefftz Duality)288

62.18．Application to Elasticity289

CHAPTER 63 Variational Inequalities and the Signorini Problem for Nonlinear Material296

63.1．Existence and Uniqueness Theorem296

63.2．Physical Motivation298

CHAPTER 64 Bifurcation for Variational Inequalities303

64.1．Basic Ideas303

64.2．Quadratic Variational Inequalities305

64.3．Lagrange Multiplier Rule for Variational Inequalities306

64.4．Main Theorem308

64.5．Proof of the Main Theorem309

64.6．Applications to the Bending of Rods and Beams311

64.7．Physical Motivation for the Nonlinear Rod Equation315

64.8．Explicit Solution of the Rod Equation317

CHAPTER 65 Pseudomonotone Operators,Bifurcation,and the von Kármán Plate Equations322

65.1．Basic Ideas322

65.2．Notations325

65.3．The von Kármán Plate Equations326

65.4．The Operator Equation327

65.5．Existence Theorem332

65.6．Bifurcation332

65.7．Physical Motivation of the Plate Equations334

65.8．Principle of Stationary Potential Energy and Plates with Obstacles339

CHAPTER 66 Convex Analysis,Maximal Monotone Operators,and Elasto-Viscoplastic Material with Linear Hardening and Hysteresis348

66.1．Abstract Model for Slow Deformation Processes349

66.2．Physical Interpretation of the Abstract Model352

66.3．Existence and Uniqueness Theorem355

66.4．Applications358

APPLICATIONS IN THERMODYNAMICS363

CHAPTER 67 Phenomenological Thermodynamics of Quasi-Equilibrium and Equilibrium States369

67.1．Thermodynamical States,Processes,and State Variables371

67.2．Gibbs'Fundamental Equation374

67.3．Applications to Gases and Liquids375

67.4．The Three Laws of Thermodynamics378

67.5．Change of Variables,Legendre Transformation,and Thermodynamical Potentials385

67.6．Extremal Principles for the Computation of Thermodynamical Equilibrium States387

67.7．Gibbs'Phase Rule391

67.8．Applications to the Law of Mass Action392

CHAPTER 68 Statistical Physics396

68.1．Basic Equations of Statistical Physics397

68.2．Bose and Fermi Statistics402

68.3．Applications to Ideal Gases403

68.4．Planck's Radiation Law408

68.5．Stefan-Boltzmann Radiation Law for Black Bodies409

68.6．The Cosmos at a Temperature of 1011K411

68.7．Basic Equation for Star Models412

68.8．Maximal Chandrasekhar Mass of White Dwarf Stars412

CHAPTER 69 Continuation with Respect to a Parameter and a Radiation Problem of Carleman422

69.1．Conservation Laws422

69.2．Basic Equations of Heat Conduction423

69.3．Existence and Uniqueness for a Heat Conduction Problem425

69.4．Proof of Theorem 69.A426

APPLICATIONS IN HYDRODYNAMICS431

CHAPTER 70 Basic Equations of Hydrodynamics433

70.1．Basic Equations434

70.2．Linear Constitutive Law for the Friction Tensor436

70.3．Applications to Viscous and Inviscid Fluids438

70.4．Tube Flows, Similarity,and Turbulence439

70.5．Physical Motivation of the Basic Equations441

70.6．Applications to Gas Dynamics444

CHAPTER 71 Bifurcation and Permanent Gravitational Waves448

71.1．Physical Problem and Complex Velocity451

71.2．Complex Flow Potential and Free Boundary-Value Problem454

71.3．Transformed Boundary-Value Problem for the Circular Ring456

71.4．Existence and Uniqueness of the Bifurcation Branch459

71.5．Proof of Theorem 71.B462

71.6．Explicit Construction of the Solution464

CHAPTER 72 Viscous Fluids and the Navier-Stokes Equations479

72.1．Basic Ideas480

72.2．Notations485

72.3．Generalized Stationary Problem486

72.4．Existence and Uniqueness Theorem for Stationary Flows490

72.5．Generalized Nonstationary Problem491

72.6．Existence and Uniqueness Theorem for Nonstationary Flows494

72.7．Taylor Problem and Bifurcation495

72.8．Proof of Theorem 72.C500

72.9．Bénard Problem and Bifurcation505

72.10．Physical Motivation of the Boussinesq Approximation512

72.11．The Kolmogorov 5/3-Law for Energy Dissipation in Turbulent Flows513

72.12．Velocity in Turbulent Flows515

MANIFOLDS AND THEIR APPLICATIONS527

CHAPTER 73 Banach Manifolds529

73.1．Local Normal Forms for Nonlinear Double Splitting Maps531

73.2．Banach Manifolds533

73.3．Strategy of the Theory of Manifolds535

73.4．Diffeomorphisms537

73.5．Tangent Space538

73.6．Tangent Map540

73.7．Higher-Order Derivatives and the Tangent Bundle541

73.8．Cotangent Bundle545

73.9．Global Solutions of Differential Equations on Manifolds and Flows546

73.10．Linearization Principle for Maps550

73.11．Two Principles for Constructing Manifolds554

73.12．Construction of Diffeomorphisms and the Generalized Morse Lemma560

73.13．Transversality563

73.14．Taylor Expansions and Jets566

73.15．Equivalence of Maps571

73.16．Multilinearization of Maps,Normal Forms,and Castastrophe Theory572

73.17．Applications to Natural Sciences579

73.18．Orientation582

73.19．Manifolds with Boundary584

73.20．Sard's Theorem587

73.21．Whitney's Embedding Theorem588

73.22．Vector Bundles589

73.23．Differentials and Derivations on Finite-Dimensional Manifolds595

CHAPTER 74 Classical Surface Theory,the Theorema Egregium of Gauss,and Differential Geometry on Manifolds609

74.1．Basic Ideas of Tensor Calculus615

74.2．Covariant and Contravariant Tensors617

74.3．Algebraic Tensor Operations621

74.4．Covariant Differentiation623

74.5．Index Principle of Mathematical Physics625

74.6．Parallel Transport and Motivation for Covariant Differentiation626

74.7．Pseudotensors and a Duality Principle627

74.8．Tensor Densities630

74.9．The Two Fundamental Forms of Gauss of Classical Surface Theory631

74.10．Metric Properties of Surfaces634

74.11．Curvature Properties of Surfaces636

74.12．Fundamental Equations and the Main Theorem of Classical Surface Theory639

74.13．Curvature Tensor and the Theorema Egregium642

74.14．Surface Maps644

74.15．Parallel Transport on Surfaces According to Levi-Civita645

74.16．Geodesics on Surfaces and a Variational Principle646

74.17．Tensor Calculus on Manifolds648

74.18．Affine Connected Manifolds649

74.19．Riemannian Manifolds651

74.20．Main Theorem About Riemannian Manifolds and the Geometric Meaning of the Curvature Tensor653

74.21．Applications to Non-Euclidean Geometry655

74.22．Strategy for a Further Development of the Differential and Integral Calculus on Manifolds663

74.23．Alternating Differentiation of Alternating Tensors664

74.24．Applications to the Calculus of Alternating Differential Forms664

74.25．Lie Derivative673

74.26．Applications to Lie Algebras of Vector Fields and Lie Groups676

CHAPTER 75 Special Theory of Relativity694

75.1．Notations699

75.2．Inertial Systems and the Postulates of the Special Theory of Relativity699

75.3．Space and Time Measurements in Inertial Systems700

75.4．Connection with Newtonian Mechanics702

75.5．Special Lorentz Transformation706

75.6．Length Contraction,Time Dilatation,and Addition Theorem for Velocities708

75.7．Lorentz Group and Poincaré Group710

75.8．Space-Time Manifold of Minkowski713

75.9．Causality and Maximal Signal Velocity714

75.10．Proper Time717

75.11．The Free Particle and the Mass-Energy Equivalence719

75.12．Energy Momentum Tensor and Relativistic Conservation Laws for Fields723

75.13．Applications to Relativistic Ideal Fluids726

CHAPTER 76 General Theory of Relativity730

76.1．Basic Equations of the General Theory of Relativity730

76.2．Motivation of the Basic Equations and the Variational Principle for the Motion of Light and Matter732

76.3．Friedman Solution for the Closed Cosmological Model736

76.4．Friedman Solution for the Open Cosmological Model741

76.5．Big Bang,Red Shift,and Expansion of the Universe742

76.6．The Future of our Cosmos745

76.7．The Very Early Cosmos747

76.8．Schwarzschild Solution756

76.9．Applications to the Motion of the Perihelion of Mercury758

76.10．Deflection of Light in the Gravitational Field of the Sun765

76.11．Red Shift in the Gravitational Field766

76.12．Virtual Singularities,Continuation of Space-Time Manifolds,and the Kruskal Solution767

76.13．Black Holes and the Sinking of a Space Ship771

76.14．White Holes775

76.15．Black-White Dipole Holes and Dual Creatures Without Radio Contact to Us775

76.16．Death of a Star776

76.17．Vaporization of Black Holes780

CHAPTER 77 Simplicial Methods,Fixed Point Theory,and Mathematical Economics794

77.1．Lemma of Sperner797

77.2．Lemma of Knaster,Kuratowski,and Mazurkiewicz798

77.3．Elementary Proof of Brouwer's Fixed-Point Theorem799

77.4．Generalized Lemma of Knaster,Kuratowski,and Mazurkiewicz800

77.5．Inequality of Fan801

77.6．Main Theorem for n-Person Games of Nash and the Minimax Theorem802

77.7．Applications to the Theorem of Hartman-Stampacchia for Variational Inequalities803

77.8．Fixed-Point Theorem of Kakutani804

77.9．Fixed-Point Theorem of Fan-Glicksberg805

77.10．Applications to the Main Theorem of Mathematical Economics About Walras Equilibria and Quasi-Variational Inequalities806

77.11．Negative Retract Principle808

77.12．Intermediate-Value Theorem of Bolzano-Poincaré-Miranda808

77.13．Equivalent Statements to Brouwer's Fixed-Point Theorem810

CHAPTER 78 Homotopy Methods and One-Dimensional Manifolds817

78.1．Basic Idea818

78.2．Regular Solution Curves818

78.3．Turning Point Principle and Bifurcation Principle821

78.4．Curve Following Algorithm822

78.5．Constructive Leray-Schauder Principle823

78.6．Constructive Approach for the Fixed-Point Index and the Mapping Degree824

78.7．Parametrized Version of Sard's Theorem828

78.8．Theorem of Sard-Smale829

78.9．Proof of Theorem 78.A830

78.10．Parametrized Version of the Theorem of Sard Smale832

78.11．Main Theorem About Generic Finiteness of the Solution Set834

78.12．Proof of Theorem 78.B834

CHAPTER 79 Dynamical Stability and Bifurcation in B-Spaces840

79.1．Asymptotic Stability and Instability of Equilibrium Points841

79.2．Proof of Theorem 79.A843

79.3．Multipliers and the Fixed-Point Trick for Dynamical Systems846

79.4．Floquet Transformation Trick848

79.5．Asymptotic Stability and Instability of Periodic Solutions851

79.6．Orbital Stability852

79.7．Perturbation of Simple Eigenvalues853

79.8．Loss of Stability and the Main Theorem About Simple Curve Bifurcation856

79.9．Loss of Stability and the Main Theorem About Hopf Bifurcation860

79.10．Proof of Theorem 79.F863

79.11．Applications to Ljapunov Bifurcation867

Appendix883

References885

List of Symbols933

List of Theorems943

List of the Most Important Definitions946

List of Basic Equations in Mathematical Physics953

Index959