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