Nonlinear Functional Analysis and its Applications

Transcription

1 Eberhard Zeidler Nonlinear Functional Analysis and its Applications IV: Applications to Mathematical Physics Translated by Juergen Quandt With 201 Illustrations Springer

2 Preface translator's Preface vii xiii INTRODUCTION Mathematics and Physics 1 APPLICATIONS IN MECHANICS 7 CHAPTER 58 Basic Equations of Point Mechanics Notations Lever Principle and Stability of the Scales Perspectives Kepler's Laws and a Look at the History of Astronomy Newton's Basic Equations Changes of the System of Reference and the Role of Inertial Systems General Point System and Its Conserved Quantities Newton's Law of Gravitation and Coulomb's Law of Electrostatics Application to the Motion of Planets Gauss' Principle of Least Constraint and the General Basic Equations of Point Mechanics with Side Conditions Principle of Virtual Power Equilibrium States and a General Stability Principle Basic Equations of the Rigid Body and the Main Theorem about the Motion of the Rigid Body and Its Equilibrium Foundation of the Basic Equations of the Rigid Body 55 xv

3 xvi Physical Models, the Expansion of the Universe, and Its Evolution after the Big Bang Legendre Transformation and Conjugate Functionals Lagrange Multipliers Principle of Stationary Action ^ Trick of Position Coordinates and Lagrangian Mechanics Hamiltonian Mechanics Poissonian Mechanics and Heisenberg's Matrix Mechanics in Quantum Theory Propagation of Action Hamilton-Jacobi Equation Canonical Transformations and the Solution of the Canonical Equations via the Hamilton-Jacobi Equation Lagrange Brackets and the Solution of the Hamilton-Jacobi Equation via the Canonical Equations Initial-Value Problem for the Hamilton-Jacobi Equation Dimension Analysis 89 CHAPTER 59 Dualism Between Wave and Particle, Preview of Quantum Theory, and Elementary Particles Plane Waves Polarization Dispersion Relations Spherical Waves Damped Oscillations and the Frequency-Time Uncertainty Relation Decay of Particles Cross Sections for Elementary Particle Processes and the Main Objectives in Quantum Field Theory Dualism Between Wave and Particle for Light Wave Packets and Group Velocity Formulation of a Particle Theory for a Classical Wave Theory Motivation of the Schrodinger Equation and Physical Intuition Fundamental Probability Interpretation of Quantum Mechanics Meaning of Eigenfunctions in Quantum Mechanics Meaning of Nonnormalized States Special Functions in Quantum Mechanics Spectrum of the Hydrogen Atom Functional Analytic Treatment of the Hydrogen Atom Harmonic Oscillator in Quantum Mechanics Heisenberg's Uncertainty Relation Pauli Principle, Spin, and Statistics Quantization of the Phase Space and Statistics Pauli Principle and the Periodic System of the Elements Classical Limiting Case of Quantum Mechanics and the WKB Method to Compute Quasi-Classical Approximations Energy-Time Uncertainty Relation and Elementary Particles The Four Fundamental Interactions Strength of the Interactions 136

4 xvii APPLICATIONS IN ELASTICITY THEORY 143 CHAPTER 60 Elastoplastic Wire Experimental Result Viscoplastic Constitutive Laws Elasto-Viscoplastic Wire with Linear Hardening Law Quasi-Statical Plasticity Some Historical Remarks on Plasticity 155 CHAPTER 61 Basic Equations of Nonlinear Elasticity Theory Notations Strain Tensor and the Geometry of Deformations Basic Equations Physical Motivation of the Basic Equations Reduced Stress Tensor and the Principle of Virtual Power A General Variational Principle (Hyperelasticity) Elastic Energy of the Cuboid and Constitutive Laws Theory of Invariants and the General Structure of Constitutive Laws and Stored Energy Functions Existence and Uniqueness in Linear Elastostatics (Generalized Solutions) Existence and Uniqueness in Linear Elastodynamics (Generalized Solutions) Strongly Elliptic Systems Local Existence and Uniqueness Theorem in Nonlinear Elasticity via the Implicit Function Theorem Existence and Uniqueness Theorem in Linear Elastostatics (Classical Solutions) Stability and Bifurcation in Nonlinear Elasticity The Continuation Method in Nonlinear Elasticity and an Approximation Method 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 Basic Ideas Notations Principle of Minimal Potential Energy, Existence, and Uniqueness Principle of Maximal Dual Energy and Duality Proofs of the Main Theorems Approximation Methods Applications to Linear Elasticity Theory Application to Nonlinear Hencky Material The Constitutive Law for Quasi-Statical Plastic Material 257

5 XV Principle of Maximal Dual Energy and the Existence Theorem for Linear Quasi-Statical Plasticity Duality and the Existence Theorem for Linear Statical Plasticity Compensated Compactness Existence Theorem for Polyconvex Material Application to Rubberlike Material Proof of Korn's Inequality Legendre Transformation and the Strategy of the General Friedrichs Duality in the Calculus of Variations Application to the Dirichlet Problem (Trefftz Duality) Application to Elasticity 289 CHAPTER 63 Variational Inequalities and the Signorini Problem for Nonlinear Material Existence and Uniqueness Theorem Physical Motivation 298 CHAPTER 64 Bifurcation for Variational Inequalities Basic Ideas Quadratic Variational Inequalities Lagrange Multiplier Rule for Variational Inequalities Main Theorem Proof of the Main Theorem Applications to the Bending of Rods and Beams Physical Motivation for the Nonlinear Rod Equation Explicit Solution of the Rod Equation 317 CHAPTER 65 Pseudomonotone Operators, Bifurcation, and the von Karman Plate Equations Basic Ideas Notations The von Karman Plate Equations The Operator Equation Existence Theorem Bifurcation Physical Motivation of the Plate Equations 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 Abstract Model for Slow Deformation Processes Physical Interpretation of the Abstract Model Existence and Uniqueness Theorem Applications 358

6 xix APPLICATIONS IN THERMODYNAMICS 363 CHAPTER 67 Phenomenological Thermodynamics of Quasi-Equilibrium and Equilibrium States Thermodynamical States, Processes, and State Variables Gibbs' Fundamental Equation Applications to Gases and Liquids The Three Laws of Thermodynamics Change of Variables, Legendre Transformation, and Thermodynamical Potentials Extremal Principles for the Computation of Thermodynamical Equilibrium States Gibbs'Phase Rule Applications to the Law of Mass Action 392 CHAPTER 68 Statistical Physics Basic Equations of Statistical Physics Bose and Fermi Statistics Applications to Ideal Gases Planck's Radiation Law Stefan-Boltzmann Radiation Law for Black Bodies The Cosmos at a Temperature of 10 1l K Basic Equation for Star Models 412 Maximal Chandrasekhar Mass of White Dwarf Stars 412 CHAPTER 69 Continuation with Respect to a Parameter and a Radiation Problem ofcarleman Conservation Laws Basic Equations of Heat Conduction Existence and Uniqueness for a Heat Conduction Problem Proof of Theorem 69. A 426 APPLICATIONS IN HYDRODYNAMICS 431 CHAPTER 70 Basic Equations of Hydrodynamics Basic Equations Linear Constitutive Law for the Friction Tensor Applications to Viscous and Inviscid Fluids Tube Flows, Similarity, and Turbulence Physical Motivation of the Basic Equations Applications to Gas Dynamics 444

7 XX CHAPTER 71 Bifurcation and Permanent Gravitational Waves Physical Problem and Complex Velocity Complex Flow Potential and Free Boundary-Value Problem Transformed Boundary-Value Problem for the Circular Ring Existence and Uniqueness of the Bifurcation Branch Proof of Theorem 71.B Explicit Construction of the Solution 464 CHAPTER 72 Viscous Fluids and the Navier-Stokes Equations Basic Ideas Notations Generalized Stationary Problem Existence and Uniqueness Theorem for Stationary Flows Generalized Nonstationary Problem Existence and Uniqueness Theorem for Nonstationary Flows Taylor Problem and Bifurcation Proof of Theorem 72.C Benard Problem and Bifurcation Physical Motivation of the Boussinesq Approximation The Kolmogorov 5/3-Law for Energy Dissipation in Turbulent Flows Velocity in Turbulent Flows 515 MANIFOLDS AND THEIR APPLICATIONS 527 CHAPTER 73 Banach Manifolds Local Normal Forms for Nonlinear Double Splitting Maps Banach Manifolds Strategy of the Theory of Manifolds Diffeomorphisms Tangent Space Tangent Map Higher-Order Derivatives and the Tangent Bundle Cotangent Bundle Global Solutions of Differential Equations on Manifolds and Flows Linearization Principle for Maps Two Principles for Constructing Manifolds Construction of Diffeomorphisms and the Generalized Morse Lemma Transversality Taylor Expansions and Jets Equivalence of Maps Multilinearization of Maps, Normal Forms, and Castastrophe Theory 572

8 xxi Applications to Natural Sciences Orientation Manifolds with Boundary Sard's Theorem Whitney's Embedding Theorem Vector Bundles Differentials and Derivations on Finite-Dimensional Manifolds 595 CHAPTER 74 Classical Surface Theory, the Theorema Egregium of Gauss, and Differential Geometry on Manifolds Basic Ideas of Tensor Calculus Co variant and Contra variant 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 676 CHAPTER 75 Special Theory of Relativity Notations Inertial Systems and the Postulates of the Special Theory of Relativity Space and Time Measurements in Inertial Systems Connection with Newtonian Mechanics Special Lorentz Transformation 706

9 xxil Length Contraction, Time Dilatation, and Addition Theorem for Velocities Lorentz Group and Poincare Group Space-Time Manifold of Minkowski Causality and Maximal Signal Velocity Proper Time The Free Particle and the Mass-Energy Equivalence Energy Momentum Tensor and Relativistic Conservation Laws for Fields Applications to Relativistic Ideal Fluids 726 CHAPTER 76 General Theory of Relativity Basic Equations of the General Theory of Relativity Motivation of the Basic Equations and the Variational Principle for the Motion of Light and Matter Friedman Solution for the Closed Cosmological Model Friedman Solution for the Open Cosmological Model Big Bang, Red Shift, and Expansion of the Universe The Future of our Cosmos The Very Early Cosmos Schwarzschild Solution Applications to the Motion of the Perihelion of Mercury Deflection of Light in the Gravitational Field of the Sun Red Shift in the Gravitational Field Virtual Singularities, Continuation of Space-Time Manifolds, and the Kruskal Solution Black Holes and the Sinking of a Space Ship White Holes Black-White Dipole Holes and Dual Creatures Without Radio Contact to Us Death of a Star Vaporization of Black Holes 780 CHAPTER 77 Simplicial Methods, Fixed Point Theory, and Mathematical Economics Lemma of Sperner Lemma of Knaster, Kuratowski, and Mazurkiewicz Elementary Proof of Brouwer's Fixed-Point Theorem Generalized Lemma of Knaster, Kuratowski, and Mazurkiewicz Inequality of Fan Main Theorem for n-person Games of Nash and the Minimax Theorem Applications to the Theorem of Hartman-Stampacchia for Variational Inequalities Fixed-Point Theorem of Kakutani Fixed-Point Theorem of Fan-Glicksberg 805

10 XX Applications to the Main Theorem of Mathematical Economics About Walras Equilibria and Quasi-Variational Inequalities Negative Retract Principle Intermediate-Value Theorem of Bolzano-Poincare-Miranda Equivalent Statements to Brouwer's Fixed-Point Theorem 810 CHAPTER 78 Homotopy Methods and One-Dimensional Manifolds Basic Idea Regular Solution Curves Turning Point Principle and Bifurcation Principle Curve Following Algorithm Constructive Leray-Schauder Principle Constructive Approach for the Fixed-Point Index and the Mapping Degree Parametrized Version of Sard's Theorem Theorem of Sard-Smale Proof of Theorem 78.A Parametrized Version of the Theorem of Sard-Smale Main Theorem About Generic Finiteness of the Solution Set Proof of Theorem 78.B 834 CHAPTER 79 Dynamical Stability and Bifurcation in B-Spaces Asymptotic Stability and Instability of Equilibrium Points Proof of Theorem 79. A Multipliers and the Fixed-Point Trick for Dynamical Systems Floquet Transformation Trick Asymptotic Stability and Instability of Periodic Solutions Orbital Stability Perturbation of Simple Eigenvalues Loss of Stability and the Main Theorem About Simple Curve Bifurcation Loss of Stability and the Main Theorem About Hopf Bifurcation Proof of Theorem 79.F 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