Topics in Nonlinear Physics

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Nonlinear Problems in Physics.- Nonlinear Problems in Physics.- The Nonlinear Field Theories in Mechanics.- 1. Thermodynamics of Homogeneous Processes.- Temperature and Heat.- Homogeneous Processes.- The Thermodynamic State.- Thermodynamic Processes.- Histories.- Thermodynamic Constitutive Equations.- Example 1. The Classical Caloric Equations of State.- 2. Kinematics. Changes of Frame.- Geometry.- Primitive Elements of Thermomechanics.- Bodies, Configurations, Motions.- Mass Density.- Reference Configuration.- Deformation Gradient.- Change of Reference Configuration.- Current Configuration as Reference.- Stretch and Rotation.- Stretching and Spin.- Changes of Frame.- 3. Force and Work, Laws of Motion.- Physical Principles of Mechanics.- Forces and Moments in Continuum Mechanics.- Momentum and Moment of Momentum.- The Euler-Cauchy Stress Principle.- Energy and Heat Work.- Equations of Balance.- Differential Equations of Continuum Mechanics.- 4. Constitutive Equations. Simple Materials.- Nature of Constitutive Equations.- Equivalent Processes.- Constitutive Equations.- Axioms for Constitutive Equations.- Simple Materials.- Exercise 4.1 (Zaremba-Noll Theorem).- Reduction for Material Frame-Indifference.- Internal Constraints.- 5. The Isotropy Group.- Isotropy Group.- Orthogonal Part of the Isotropy Group.- Change of Reference Configuration.- 6. Solids, Isotropic Materials, Fluids, Fluid Crystals.- Main Properties of the Isotropy Group.- Isotropic Materials.- Solids.- Fluids.- Fluid Crystals.- 7. Motions with Constant Stretch History.- Definition.- Noll’s Theorem.- Determination of the Deformation History from the First Three Rivlin-Ericksen Tensors.- Equivalence of Simple Materials with Rivlin-Ericksen Materials in Motions with Constant Stretch History.- Classification of Motions with Constant Stretch History.- 8. The Stress System in Viscometric Flows of Incompressible Fluids.- Recapitulation.- Functional Equations for the Response Function.- Illustration by Means of Shearing Flow.- Consequences of Invariance Under Reflections.- The Viscometric Functions. Normal-Stress Effects.- Position of the Classical Theory.- 9. Dynamical Conditions in Viscometric Flows.- Recapitulation.- Dynamic Compatibility.- Shearing Flow.- Channel Flow.- Helical Flows.- Flow Between Rotating Cylinders.- Flow in a Circular Pipe.- Other Viscometric Flows.- 10. Impossibility of Rectilinear Plow in Pipes.- Problem.- Explicit Constitutive Equation.- Dynamical Equation.- Compatibility.- 11. Elastic Materials.- Statics of Simple Materials.- Reduction for Frame-Indifference.- Elastic Fluids.- Natural States.- Isotropic Elastic Materials.- Unconstrained Bodies.- Incompressible Bodies.- Remarks.- 12. Non-Homogeneous Universal Solutions for Incompressible Elastic Bodies.- Universal Solutions.- List of the Universal Solutions.- Inflation or Eversion, Torsion, and Extension of a Cylinder.- Torsion and Tension of a Solid Cylinder.- Eversion.- Remarks on Method.- 13. Hyperelastic Materials.- The Piola-Kirchhoff Stress Tensor.- Definition of a Hyperelastic Material.- The Two Isotropy Groups.- Minimum of the Stored-Energy Function.- Need for an A Priori Inequality.- Coleman and Noll’s Inequality.- 14. Work Theorems in Hyperelasticity.- Nonsense About Perpetual Motion.- Virtual Work of the Traction on the Boundary.- Homogeneous Processes in Homogeneous Bodies.- The First Work Theorem.- The Second Work Theorem.- The Third Work Theorem.- 15. Thermomechanics. Equipresence.- Fading Memory.- Thermodynamics and Continuum Mechanics Reviewed.- General Principles of the Thermomechanics of Continua.- Thermodynamic Process.- Constitutive Equations. Equipresence.- The Clausius-Duhem Inequality.- The Reduced Dissipation Inequality.- Simple Thermodynamic Materials.- 16. Thermodynamics of Simple Materials with Fading Memory.- Quasi-Elastic Response. Fading Memory.- Dissipation and Quasi-Elastic Response.- Applications.- A Final Remark on the Stored-Energy Function in Elasticity.- 17. Wave Motions. Compatibility.- The Nature of Wave Motions.- Hadamard’s Lemma.- Singular Surfaces.- Singular Surfaces for a Motion.- Compatibility Conditions for Second-Order Singular Surfaces.- Equation of Balance at a Singular Surface.- Acceleration Waves in Elasticity.- Weak Singular Surfaces in General.- 18. Wave Propagation in Dissipative Materials.- The “Smoothing” Effect of Dissipation.- Waves and Quasi-Elastic Response.- Relation Between the Homo-thermal and Homentropic Acoustic Tensors.- Waves in Non-Conductors.- Waves in Definite Conductors.- Closure.- to Nonequilibrium Statistical Physics.- 1. Introduction.- References.- 2. Phenomenological Approach — Thermodynamics of Irreversible Processes.- References.- 3. Statistical Mechanics — General Method.- a) Introduction.- b) Liouville Equation.- c) Free Particles.- d) Potential Scattering.- e) General Theory.- f) General Evolution Equation.- g) Transformation Theory of the Kinetic Equation.- h) Correlations.- References.- 4. Boltzmann Situations.- a) Anharmonlc Solids.- b) Brownlan Motion.- References.- 5. Generalized Boltzmann Situations.- a) The Three-Body Problem.- b) Strongly Coupled Systems.- c) The RenormallzatIon Program.- Strongly Coupled Anharmonlc Oscillators.- d) Plön Production in Baryon-Pion Scattering with a p-Meson in the Intermediate Stage.- e) Relation with S-Matrix Theory.- References.- 6. NonBoltzmann Situations.- a) General Remarks.- b) Harmonic Lattices.- c) Heisenberg Spin Systems.- d) Gravitational Plasmas.- References.- 7. Irreversibility.- a) Introduction.- b) Discussion of Zermelo’s Paradox.- c) Discussion of Loschmidt’s Paradox.- d) Meaning of Irreversibility.- References.- 8. Entropy.- a) Thermodynamic Entropy and Statistical Mechanics.- b) Entropy in Strongly Coupled Anharmonic Oscillators -Disorder and Entropy.- c) Canonical Entropy and Resonance Effects.- d) Numerical Calculations.- References.- 9. Concluding Remarks.- References.- Appendix I. The General Evolution Equation and Projection Operators.- References.- Appendix II. Nonanalytic Density Behavior of Transport Coefficients.- References.- Interactions in a Classical Relativistic Plasma.- Preface.- 1. Introduction.- 2. Lorentz-Invariant Statistical Mechanical Formalism for a Classical Relativistlc Plasma Interacting with an Electromagnetic Field.- a) Introduction.- b) Lorentz-Invariant Formalism.- c) Perturbation Treatment of the Liouville Equation.- 3. Formal Approach to the Phenomenological Electrodynamics of Plasmas.- 4. Kinetic Approach to the Electrodynamics of Homogeneous Systems.- a) Kinetic Equations.- b) General Description of a Plasma Within the Ring Approximation.- b.1 Self-energy.- b.2 Relativistic Landau Equation.- b.3 General Ring Equations.- b.4 Ring Equations for Stable Systems.- b.5 Ring Equations for Unstable Systems.- c) Typical Radiation Processes.- c.1 Thomson Scattering.- c.2 Normal and Anomalous BremsStrahlung.- 5. Kinetic Approach to Phenomenological Radiation Laws for Homogeneous Systems.- a) General Radiation Transfer Equation.- b) Relation Between Absorption Coefficient and Electrical Conductivity.- References.- Nonlinear Optics.- 1. Phenomenological Survey of Nonlinear Optical Effects.- 1.1 The Anharmonic Oscillator as a Classical Model of Nonlinearity.- 1.2 Nonlinear Source Terms.- 1.3 Nonlinearities in Media with Inversion Symmetry and in Isotropic Media.- 2. Wave Propagation in Nonlinear Media.- 2.1 The Nonlinear Wave Equation.- 2.2 Bogoliubov’s Method for a One-dimensional Nonlinear Mechanical System.- 2.3 The Secular and Nonsecular Behavior of the Wave Equation.- 3. Boundary Conditions and Coupling Between Light Waves in a Dispersive Medium.- 3.1 Parametric Generation.- 3.2 Secular Harmonic Conversion.- 3.3 The Propagation of a Wave in a Medium with a Complex Intensity Dependent Index of Refraction.- 3.4 Saturable Absorber.- 3.5 Second Harmonic Generation in Dissipative Crystals Without Inversion Symmetry.- 4. Stimulated Brillouin, Raman and Rayleigh Scattering Effects.- 4.1 Self-Focusing of the Laser Beam.- 4.2 Parametric Down Conversion.- 4.3 The Stimulated Raman Scattering as a Parametric Process.- 5. Quantum Mechanical Calculation of Nonlinear Susceptibilities. Lamb’s Theory of Coupled Laser Modes.- 5.1 The Coupling of One Electromagnetic Mode with Independent Particles with Two Energy Levels.- 5.2 Quantum Mechanical Calculation of Nonlinear Susceptibilities.- 5.3 Oscillation in a Single Mode.- 5.4 Two Oscillating Modes.- 5.5 Three Coupled Modes in a Gas Laser.- Lectures on Homogeneous Turbulence.- Preface.- 1. Introduction.- Diffusive Property of Turbulence.- Strong Nonlinearity.- Intermittency.- Homogeneous Turbulence.- Appendix 1..- 2. The Description of Homogeneous Turbulence.- Correlation Tensors.- Spectral Tensors.- The Probability Distribution.- Formulation of the Problem of Homogeneous Turbulence.- Isotropic Turbulence.- Do All Mean Values Exist?.- 3. Structure and Invariance of the Large Eddies.- Fourier Analysis of the Velocity Field.- Form of the Energy Spectrum Tensor for Small Wavenumber.- The Form of the Velocity Correlation Tensor at Large Separation.- The Case C=0.- Stationary Homogeneous Turbulence.- 4. The Problem of Decay.- The Generation of Vorticity.- 5. The Universal Equilibrium Theory (Kolmogorov).- Deductions.- Inertial Subrange.- Experimental Evidence.- Alternative Explanation of the Dependence on ? and v and the k-5/3 Dependence.- 6. “Burgerlence”.- Decay of a Single Pulse.- Decay of a Periodic Train of Pulses.- Decay of a General Disturbance.- The Spectrum Function.- Simple Derivation of the Qualitative Dependence.- The Kolmogorov Length in Burgerlence.- 7. The Structure of Turbulence.- Two Exact Solutions of the Navier-Stokes Equations.- Random Collections of Sheets and Tubes.- A Model of Turbulent Motion.- Predictions of the Model.- Remarks.- 8. Quasianalytical Theories of Turbulence.- Theories Based on Physical Approximations.- The Joint Normal Approximation or Zero Fourth-Order Cumulant Hypothesis.- The Wiener-Hermite Expansion.- The Kraichnan Direct Interaction Approximation.- 9. Concluding Remarks.- References.- Superspace and the Nature of Quantum Geometrodynamics.- Allowable History Selected out of Arena of Dynamics by Constructive Interference.- Three Dimensions, Not Four.- Superspace.- Wave Packet in Superspace and its Propagation.- “Spacetime” a Concept of Limited Validity.- The Planck Length and Gravitational Collapse.- Quantum Fluctuations in Geometry of Space.- Fluctuations Superposed on Classical Background.- Extrapolate Geometrodynamics to the Planck Scales of Distances?.- Electricity as Lines of Force Trapped in the Topology of Space.- The Energy of the Vacuum.- A Particle as a Geometrodynamical Exciton.- Problem 1. “Derivation” of “Einstein-Hamilton-Jacobi Equation”.- Subscript on Relation of Hamilton-Jacobi Method to Conventional Analytic Solutions of Field Equations.- Problem 2. Structure of Superspace.- Level 1. Classical Geometrodynamics; Topology Does Not Change.- Fixed Topology Excludes GMD Account of Particles and Fields.- Formalism of Field When Treated as “Foreign and Physical”.- Level 2. Space Resonating Between 3-Geometries of Varied Topology.- The Orientation Entanglement Relation or “Version”.- Alternative Spin Structures in a Multiply Connected Space.- The Multisheeted Character of Superspace.- Electromagnetlsm as a Statistical Aspect of Geometry? Other Questions.- The Example of 2-Geometries.- Other Aspects of Superspace.- Tangent Vectors on Superspace and the Classical Initial Value Problem.- Level 3. Pregeometry.- Problem 3. Initial Conditions.- References.