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电子书下载

I NTRODUCTION Mathematics and Physics 1

APPLICATIONS IN MECHANICS 7

CHAPTER 58 Basic Equations of Point Mechanics 9

58.1．Notations 10

58.2．Lever Principle and Stability of the Scales 14

58.3．Perspectives 17

58.4．Kepler's Laws and a Look at the History of Astronomy 22

58.5．Newton's Basic Equations 25

58.6．Changes of the System of Reference and the Role of Inertial Systems 28

58.7．General Point System and Its Conserved Quantities 32

58.8．Newton's Law of Gravitation and Coulomb's Law of Electrostatics 35

58.9．Application to the Motion of Planets 38

58.10．Gauss'Principle of Least Constraint and the General Basic Equations of Point Mechanics with Side Conditions 45

58.11．Principle of Virtual Power 48

58.12．Equilibrium States and a General Stability Principle 50

58.13．Basic Equations of the Rigid Body and the Main Theorem about the Motion of the Rigid Body and Its Equilibrium 52

58.14．Foundation of the Basic Equations of the Rigid Body 55

58.15．Physical Models,the Expansion of the Universe,and Its Evolution after the Big Bang 57

58.16．Legendre Transformation and Conjugate Functionals 65

58.17．Lagrange Multipliers 67

58.18．Principle of Stationary Action 69

58.19．Trick of Position Coordinates and Lagrangian Mechanics 70

58.20．Hamiltonian Mechanics 72

58.21．Poissonian Mechanics and Heisenberg's Matrix Mechanics in Quantum Theory 77

58.22．Propagation of Action 81

58.23．Hamilton-Jacobi Equation 82

58.24．Canonical Transformations and the Solution of the Canonical Equations via the Hamilton-Jacobi Equation 83

58.25．Lagrange Brackets and the Solution of the Hamilton-Jacobi Equation via the Canonical Equations 84

58.26．Initial-Value Problem for the Hamilton Jacobi Equation 87

58.27．Dimension Analysis 89

CHAPTER 59 Dualism Between Wave and Particle,Preview of Quantum Theory,and Elementary Particles 98

59.1．Plane Waves 99

59.2．Polarization 101

59.3．Dispersion Relations 102

59.4．Spherical Waves 103

59.5．Damped Oscillations and the Frequency-Time Uncertainty Relation 104

59.6．Decay of Particles 105

59.7．Cross Sections for Elementary Particle Processes and the Main Objectives in Quantum Field Theory 106

59.8．Dualism Between Wave and Particle for Light 107

59.9．Wave Packets and Group Velocity 110

59.10．Formulation of a Particle Theory for a Classical Wave Theory 111

59.11．Motivation of the Schr？dinger Equation and Physical Intuition 112

59.12．Fundamental Probability Interpretation of Quantum Mechanics 113

59.13．Meaning of Eigenfunctions in Quantum Mechanics 114

59.14．Meaning of Nonnormalized States 116

59.15．Special Functions in Quantum Mechanics 117

59.16．Spectrum of the Hydrogen Atom 118

59.17．Functional Analytic Treatment of the Hydrogen Atom 121

59.18．Harmonic Oscillator in Quantum Mechanics 122

59.19．Heisenberg's Uncertainty Relation 123

59.20．Pauli Principle,Spin,and Statistics 125

59.21．Quantization of the Phase Space and Statistics 126

59.22．Pauli Principle and the Periodic System of the Elements 127

59.23．Classical Limiting Case of Quantum Mechanics and the WKB Method to Compute Quasi-Classical Approximations 129

59.24．Energy-Time Uncertainty Relation and Elementary Particles 130

59.25．The Four Fundamental Interactions 134

59.26．Strength of the Interactions 136

APPLICATIONS IN ELASTICITY THEORY 143

CHAPTER 60 Elastoplastic Wire 145

60.1．Experimental Result 147

60.2．Viscoplastic Constitutive Laws 149

60.3．Elasto-Viscoplastic Wire with Linear Hardening Law 151

60.4．Quasi-Statical Plasticity 154

60.5．Some Historical Remarks on Plasticity 155

CHAPTER 61 Basic Equations of Nonlinear Elasticity Theory 158

61.1．Notations 166

61.2．Strain Tensor and the Geometry of Deformations 168

61.3．Basic Equations 176

61.4．Physical Motivation of the Basic Equations 180

61.5．Reduced Stress Tensor and the Principle of Virtual Power 184

61.6．A General Variational Principle (Hyperelasticity) 190

61.7．Elastic Energy of the Cuboid and Constitutive Laws 198

61.8．Theory of Invariants and the General Structure of Constitutive Laws and Stored Energy Functions 202

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 Systems 213

61.12．Local Existence and Uniqueness Theorem in Nonlinear Elasticity via the Implicit Function Theorem 215

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

61.14．Stability and Bifurcation in Nonlinear Elasticity 221

61.15．The Continuation Method in Nonlinear Elasticity and an Approximation Method 224

61.16．Convergence of the Approximation Method 227

CHAPTER 62 Monotone Potential Operators and a Class of Models with Nonlinear Hooke's Law,Duality and Plasticity,and Polyconvexity 233

62.1．Basic Ideas 234

62.2．Notations 242

62.3．Principle of Minimal Potential Energy,Existence,and Uniqueness 244

62.4．Principle of Maximal Dual Energy and Duality 245

62.5．Proofs of the Main Theorems 247

62.6．Approximation Methods 252

62.7．Applications to Linear Elasticity Theory 255

62.8．Application to Nonlinear Hencky Material 256

62.9．The Constitutive Law for Quasi-Statical Plastic Material 257

62.10．Principle of Maximal Dual Energy and the Existence Theorem for Linear Quasi-Statical Plasticity 259

62.11．Duality and the Existence Theorem for Linear Statical Plasticity 262

62.12．Compensated Compactness 264

62.13．Existence Theorem for Polyconvex Material 273

62.14．Application to Rubberlike Material 277

62.15．Proof of Korn's Inequality 278

62.16．Legendre Transformation and the Strategy of the General Friedrichs Duality in the Calculus of Variations 284

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

62.18．Application to Elasticity 289

CHAPTER 63 Variational Inequalities and the Signorini Problem for Nonlinear Material 296

63.1．Existence and Uniqueness Theorem 296

63.2．Physical Motivation 298

CHAPTER 64 Bifurcation for Variational Inequalities 303

64.1．Basic Ideas 303

64.2．Quadratic Variational Inequalities 305

64.3．Lagrange Multiplier Rule for Variational Inequalities 306

64.4．Main Theorem 308

64.5．Proof of the Main Theorem 309

64.6．Applications to the Bending of Rods and Beams 311

64.7．Physical Motivation for the Nonlinear Rod Equation 315

64.8．Explicit Solution of the Rod Equation 317

CHAPTER 65 Pseudomonotone Operators,Bifurcation,and the von Kármán Plate Equations 322

65.1．Basic Ideas 322

65.2．Notations 325

65.3．The von Kármán Plate Equations 326

65.4．The Operator Equation 327

65.5．Existence Theorem 332

65.6．Bifurcation 332

65.7．Physical Motivation of the Plate Equations 334

65.8．Principle of Stationary Potential Energy and Plates with Obstacles 339

CHAPTER 66 Convex Analysis,Maximal Monotone Operators,and Elasto-Viscoplastic Material with Linear Hardening and Hysteresis 348

66.1．Abstract Model for Slow Deformation Processes 349

66.2．Physical Interpretation of the Abstract Model 352

66.3．Existence and Uniqueness Theorem 355

66.4．Applications 358

APPLICATIONS IN THERMODYNAMICS 363

CHAPTER 67 Phenomenological Thermodynamics of Quasi-Equilibrium and Equilibrium States 369

67.1．Thermodynamical States,Processes,and State Variables 371

67.2．Gibbs'Fundamental Equation 374

67.3．Applications to Gases and Liquids 375

67.4．The Three Laws of Thermodynamics 378

67.5．Change of Variables,Legendre Transformation,and Thermodynamical Potentials 385

67.6．Extremal Principles for the Computation of Thermodynamical Equilibrium States 387

67.7．Gibbs'Phase Rule 391

67.8．Applications to the Law of Mass Action 392

CHAPTER 68 Statistical Physics 396

68.1．Basic Equations of Statistical Physics 397

68.2．Bose and Fermi Statistics 402

68.3．Applications to Ideal Gases 403

68.4．Planck's Radiation Law 408

68.5．Stefan-Boltzmann Radiation Law for Black Bodies 409

68.6．The Cosmos at a Temperature of 1011K 411

68.7．Basic Equation for Star Models 412

68.8．Maximal Chandrasekhar Mass of White Dwarf Stars 412

CHAPTER 69 Continuation with Respect to a Parameter and a Radiation Problem of Carleman 422

69.1．Conservation Laws 422

69.2．Basic Equations of Heat Conduction 423

69.3．Existence and Uniqueness for a Heat Conduction Problem 425

69.4．Proof of Theorem 69.A 426

APPLICATIONS IN HYDRODYNAMICS 431

CHAPTER 70 Basic Equations of Hydrodynamics 433

70.1．Basic Equations 434

70.2．Linear Constitutive Law for the Friction Tensor 436

70.3．Applications to Viscous and Inviscid Fluids 438

70.4．Tube Flows, Similarity,and Turbulence 439

70.5．Physical Motivation of the Basic Equations 441

70.6．Applications to Gas Dynamics 444

CHAPTER 71 Bifurcation and Permanent Gravitational Waves 448

71.1．Physical Problem and Complex Velocity 451

71.2．Complex Flow Potential and Free Boundary-Value Problem 454

71.3．Transformed Boundary-Value Problem for the Circular Ring 456

71.4．Existence and Uniqueness of the Bifurcation Branch 459

71.5．Proof of Theorem 71.B 462

71.6．Explicit Construction of the Solution 464

CHAPTER 72 Viscous Fluids and the Navier-Stokes Equations 479

72.1．Basic Ideas 480

72.2．Notations 485

72.3．Generalized Stationary Problem 486

72.4．Existence and Uniqueness Theorem for Stationary Flows 490

72.5．Generalized Nonstationary Problem 491

72.6．Existence and Uniqueness Theorem for Nonstationary Flows 494

72.7．Taylor Problem and Bifurcation 495

72.8．Proof of Theorem 72.C 500

72.9．Bénard Problem and Bifurcation 505

72.10．Physical Motivation of the Boussinesq Approximation 512

72.11．The Kolmogorov 5/3-Law for Energy Dissipation in Turbulent Flows 513

72.12．Velocity in Turbulent Flows 515

MANIFOLDS AND THEIR APPLICATIONS 527

CHAPTER 73 Banach Manifolds 529

73.1．Local Normal Forms for Nonlinear Double Splitting Maps 531

73.2．Banach Manifolds 533

73.3．Strategy of the Theory of Manifolds 535

73.4．Diffeomorphisms 537

73.5．Tangent Space 538

73.6．Tangent Map 540

73.7．Higher-Order Derivatives and the Tangent Bundle 541

73.8．Cotangent Bundle 545

73.9．Global Solutions of Differential Equations on Manifolds and Flows 546

73.10．Linearization Principle for Maps 550

73.11．Two Principles for Constructing Manifolds 554

73.12．Construction of Diffeomorphisms and the Generalized Morse Lemma 560

73.13．Transversality 563

73.14．Taylor Expansions and Jets 566

73.15．Equivalence of Maps 571

73.16．Multilinearization of Maps,Normal Forms,and Castastrophe Theory 572

73.17．Applications to Natural Sciences 579

73.18．Orientation 582

73.19．Manifolds with Boundary 584

73.20．Sard's Theorem 587

73.21．Whitney's Embedding Theorem 588

73.22．Vector Bundles 589

73.23．Differentials and Derivations on Finite-Dimensional Manifolds 595

CHAPTER 74 Classical Surface Theory,the Theorema Egregium of Gauss,and Differential Geometry on Manifolds 609

74.1．Basic Ideas of Tensor Calculus 615

74.2．Covariant and Contravariant Tensors 617

74.3．Algebraic Tensor Operations 621

74.4．Covariant Differentiation 623

74.5．Index Principle of Mathematical Physics 625

74.6．Parallel Transport and Motivation for Covariant Differentiation 626

74.7．Pseudotensors and a Duality Principle 627

74.8．Tensor Densities 630

74.9．The Two Fundamental Forms of Gauss of Classical Surface Theory 631

74.10．Metric Properties of Surfaces 634

74.11．Curvature Properties of Surfaces 636

74.12．Fundamental Equations and the Main Theorem of Classical Surface Theory 639

74.13．Curvature Tensor and the Theorema Egregium 642

74.14．Surface Maps 644

74.15．Parallel Transport on Surfaces According to Levi-Civita 645

74.16．Geodesics on Surfaces and a Variational Principle 646

74.17．Tensor Calculus on Manifolds 648

74.18．Affine Connected Manifolds 649

74.19．Riemannian Manifolds 651

74.20．Main Theorem About Riemannian Manifolds and the Geometric Meaning of the Curvature Tensor 653

74.21．Applications to Non-Euclidean Geometry 655

74.22．Strategy for a Further Development of the Differential and Integral Calculus on Manifolds 663

74.23．Alternating Differentiation of Alternating Tensors 664

74.24．Applications to the Calculus of Alternating Differential Forms 664

74.25．Lie Derivative 673

74.26．Applications to Lie Algebras of Vector Fields and Lie Groups 676

CHAPTER 75 Special Theory of Relativity 694

75.1．Notations 699

75.2．Inertial Systems and the Postulates of the Special Theory of Relativity 699

75.3．Space and Time Measurements in Inertial Systems 700

75.4．Connection with Newtonian Mechanics 702

75.5．Special Lorentz Transformation 706

75.6．Length Contraction,Time Dilatation,and Addition Theorem for Velocities 708

75.7．Lorentz Group and Poincaré Group 710

75.8．Space-Time Manifold of Minkowski 713

75.9．Causality and Maximal Signal Velocity 714

75.10．Proper Time 717

75.11．The Free Particle and the Mass-Energy Equivalence 719

75.12．Energy Momentum Tensor and Relativistic Conservation Laws for Fields 723

75.13．Applications to Relativistic Ideal Fluids 726

CHAPTER 76 General Theory of Relativity 730

76.1．Basic Equations of the General Theory of Relativity 730

76.2．Motivation of the Basic Equations and the Variational Principle for the Motion of Light and Matter 732

76.3．Friedman Solution for the Closed Cosmological Model 736

76.4．Friedman Solution for the Open Cosmological Model 741

76.5．Big Bang,Red Shift,and Expansion of the Universe 742

76.6．The Future of our Cosmos 745

76.7．The Very Early Cosmos 747

76.8．Schwarzschild Solution 756

76.9．Applications to the Motion of the Perihelion of Mercury 758

76.10．Deflection of Light in the Gravitational Field of the Sun 765

76.11．Red Shift in the Gravitational Field 766

76.12．Virtual Singularities,Continuation of Space-Time Manifolds,and the Kruskal Solution 767

76.13．Black Holes and the Sinking of a Space Ship 771

76.14．White Holes 775

76.15．Black-White Dipole Holes and Dual Creatures Without Radio Contact to Us 775

76.16．Death of a Star 776

76.17．Vaporization of Black Holes 780

CHAPTER 77 Simplicial Methods,Fixed Point Theory,and Mathematical Economics 794

77.1．Lemma of Sperner 797

77.2．Lemma of Knaster,Kuratowski,and Mazurkiewicz 798

77.3．Elementary Proof of Brouwer's Fixed-Point Theorem 799

77.4．Generalized Lemma of Knaster,Kuratowski,and Mazurkiewicz 800

77.5．Inequality of Fan 801

77.6．Main Theorem for n-Person Games of Nash and the Minimax Theorem 802

77.7．Applications to the Theorem of Hartman-Stampacchia for Variational Inequalities 803

77.8．Fixed-Point Theorem of Kakutani 804

77.9．Fixed-Point Theorem of Fan-Glicksberg 805

77.10．Applications to the Main Theorem of Mathematical Economics About Walras Equilibria and Quasi-Variational Inequalities 806

77.11．Negative Retract Principle 808

77.12．Intermediate-Value Theorem of Bolzano-Poincaré-Miranda 808

77.13．Equivalent Statements to Brouwer's Fixed-Point Theorem 810

CHAPTER 78 Homotopy Methods and One-Dimensional Manifolds 817

78.1．Basic Idea 818

78.2．Regular Solution Curves 818

78.3．Turning Point Principle and Bifurcation Principle 821

78.4．Curve Following Algorithm 822

78.5．Constructive Leray-Schauder Principle 823

78.6．Constructive Approach for the Fixed-Point Index and the Mapping Degree 824

78.7．Parametrized Version of Sard's Theorem 828

78.8．Theorem of Sard-Smale 829

78.9．Proof of Theorem 78.A 830

78.10．Parametrized Version of the Theorem of Sard Smale 832

78.11．Main Theorem About Generic Finiteness of the Solution Set 834

78.12．Proof of Theorem 78.B 834

CHAPTER 79 Dynamical Stability and Bifurcation in B-Spaces 840

79.1．Asymptotic Stability and Instability of Equilibrium Points 841

79.2．Proof of Theorem 79.A 843

79.3．Multipliers and the Fixed-Point Trick for Dynamical Systems 846

79.4．Floquet Transformation Trick 848

79.5．Asymptotic Stability and Instability of Periodic Solutions 851

79.6．Orbital Stability 852

79.7．Perturbation of Simple Eigenvalues 853

79.8．Loss of Stability and the Main Theorem About Simple Curve Bifurcation 856

79.9．Loss of Stability and the Main Theorem About Hopf Bifurcation 860

79.10．Proof of Theorem 79.F 863

79.11．Applications to Ljapunov Bifurcation 867

Appendix 883

References 885

List of Symbols 933

List of Theorems 943

List of the Most Important Definitions 946

List of Basic Equations in Mathematical Physics 953

Index 959