Eberhard JZeidler

Nonlinear Functional Analysis and its Applications

IV: Applications to Mathematical Physics

Translated by Juergen Quandt

With 201 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

Contents

Preface vii Translator's Preface xiii

INTRODUCTION

Mathematics and Physics 1

A P P L I C A T I O N S I N M E C H A N I C S 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

xv

XVI Contents

§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

CHAP T E R 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 Schrodinger 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

Contents XV11

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

XV111 Contents

§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

C H A P T E R 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 Kärmän Plate Equations 322 §65.1. Basic Ideas 322 §65.2. Notations 325 §65.3. The von Kärmän 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

CH AP T E R 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

Contents XIX

A P P L I C A T I O N S I N T H E R M O D Y N A M I C S 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 101: К 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

XX Contents

C HAP T E R 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 7 LB 462 §71.6. Explicit Construction of the Solution 464

C HAP T E R 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. Benard 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

Contents XXI

§73.17. §73.18. §73.19. §73.20. §73.21. §73.22. §73.23.

Applications to Natural Sciences Orientation Manifolds with Boundary Sard's Theorem Whitney's Embedding Theorem Vector Bundles Differentials and Derivations on Finite-Dimensional Manifolds

579 582 584 587 588 589 595

CHAPTER 74 Classical Surface Theory, the Theorema Egregium of Gauss, and Differential Geometry on Manifolds §74.1. §74.2. §74.3. §74.4. §74.5. §74.6. §74.7. §74.8. §74.9.

§74.10. §74.11. §74.12.

§74.13. §74.14. §74.15. §74.16. §74.17. §74.18. §74.19. §74.20.

§74.21. §74.22.

§74.23. §74.24. §74.25. §74.26.

Basic Ideas of Tensor Calculus Covariant and Contravariant Tensors Algebraic Tensor Operations Covariant Differentiation Index Principle of Mathematical Physics Parallel Transport and Motivation for Covariant Differentiation Pseudotensors and a Duality Principle Tensor Densities The Two Fundamental Forms of Gauss of Classical Surface Theory Metric Properties of Surfaces Curvature Properties of Surfaces Fundamental Equations and the Main Theorem of Classical Surface Theory Curvature Tensor and the Theorema Egregium Surface Maps Parallel Transport on Surfaces According to Levi-Civita Geodesies on Surfaces and a Variational Principle Tensor Calculus on Manifolds Affine Connected Manifolds Riemannian Manifolds Main Theorem About Riemannian Manifolds and the Geometric Meaning of the Curvature Tensor Applications to Non-Euclidean Geometry Strategy for a Further Development of the Differential and Integral Calculus on Manifolds Alternating Differentiation of Alternating Tensors Applications to the Calculus of Alternating Differential Forms Lie Derivative Applications to Lie Algebras of Vector Fields and Lie Groups

609 615 617 621 623 625 626 627 630 631 634 636

639 642 644 645 646 648 649 651

653 655

663 664 664 673 676

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

XXII Contents

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

§75.7. Lorentz Group and Poincare 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

CHAP T E R 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

Contents ххш

§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-Poincare-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