An Introduction to Mathematical Modeling: A Course in Mechanics

A modern approach to mathematical modeling, featuring unique applications from the field of mechanics An Introduction to Mathematical Modeling: A Course in Mechanics is designed to survey the mathematical models that form the foundations of modern science and incorporates examples that illustrate how the most successful models arise from basic principles in modern and classical mathematical physics. Written by a world authority on mathematical theory and computational mechanics, the book presents an account of continuum mechanics, electromagnetic field theory, quantum mechanics, and statistical mechanics for readers with varied backgrounds in engineering, computer science, mathematics, and physics. The author streamlines a comprehensive understanding of the topic in three clearly organized sections: Nonlinear Continuum Mechanics introduces kinematics as well as force and stress in deformable bodies; mass and momentum; balance of linear and angular momentum; conservation of energy; and constitutive equations Electromagnetic Field Theory and Quantum Mechanics contains a brief account of electromagnetic wave theory and Maxwell's equations as well as an introductory account of quantum mechanics with related topics including ab initio methods and Spin and Pauli's principles Statistical Mechanics presents an introduction to statistical mechanics of systems in thermodynamic equilibrium as well as continuum mechanics, quantum mechanics, and molecular dynamics Each part of the book concludes with exercise sets that allow readers to test their understanding of the presented material. Key theorems and fundamental equations are highlighted throughout, and an extensive bibliography outlines resources for further study. Extensively class-tested to ensure an accessible presentation, An Introduction to Mathematical Modeling is an excellent book for courses on introductory mathematical modeling and statistical mechanics at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering.

Table of Contents:
Preface xiii I Nonlinear Continuum Mechanics 1 1 Kinematics of Deformable Bodies 3 1.1 Motion 4 1.2 Strain and Deformation Tensors 7 1.3 Rates of Motion 10 1.4 Rates of Deformation 13 1.5 The Piola Transformation 15 1.6 The Polar Decomposition Theorem 19 1.7 Principal Directions and Invariants of Deformation and Strain 20 1.8 The Reynolds' Transport Theorem 23 2 Mass and Momentum 25 2.1 Local Forms of the Principle of Conservation of Mass 26 2.2 Momentum 28 3 Force and Stress in Deformable Bodies 29 4 The Principles of Balance of Linear and Angular Momentum 35 4.1 Cauchy's Theorem: The Cauchy Stress Tensor 36 4.2 The Equations of Motion (Linear Momentum) 38 4.3 The Equations of Motion Referred to the Reference Configuration: The Piola-Kirchhoff Stress Tensors 40 4.4 Power 42 5 The Principle of Conservation of Energy 45 5.1 Energy and the Conservation of Energy 45 5.2 Local Forms of the Principle of Conservation of Energy 47 6 Thermodynamics of Continua and the Second Law 49 7 Constitutive Equations 53 7.1 Rules and Principles for Constitutive Equations 54 7.2 Principle of Material Frame Indifference 57 7.2.1 Solids 57 7.2.2 Fluids 59 7.3 The Coleman-Noll Method: Consistency with the Second Law of Thermodynamics 60 8 Examples and Applications 63 8.1 The Navier-Stokes Equations for Incompressible Flow 63 8.2 Flow of Gases and Compressible Fluids: The Compressible Navier-Stokes Equations 66 8.3 Heat Conduction 67 8.4 Theory of Elasticity 69 II Electromagnetic Field Theory and Quantum Mechanics 73 9 Electromagnetic Waves 75 9.1 Introduction 75 9.2 Electric Fields 75 9.3 Gauss's Law 79 9.4 Electric Potential Energy 80 9.4.1 Atom Models 80 9.5 Magnetic Fields 81 9.6 Some Properties of Waves 84 9.7 Maxwell's Equations 87 9.8 Electromagnetic Waves 91 10 Introduction to Quantum Mechanics 93 10.1 Introductory Comments 93 10.2 Wave and Particle Mechanics 94 10.3 Heisenberg's Uncertainty Principle 97 10.4 Schrödinger's Equation 99 10.4.1 The Case of a Free Particle 99 10.4.2 Superposition in Rn 101 10.4.3 Hamiltonian Form 102 10.4.4 The Case of Potential Energy 102 10.4.5 Relativistic Quantum Mechanics 102 10.4.6 General Formulations of Schrödinger's Equation 103 10.4.7 The Time-Independent Schrödinger Equation 104 10.5 Elementary Properties of the Wave Equation 104 10.5.1 Review 104 10.5.2 Momentum 106 10.5.3 Wave Packets and Fourier Transforms 109 10.6 The Wave-Momentum Duality 110 10.7 Appendix: A Brief Review of Probability Densities 111 11 Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism 115 11.1 Introductory Remarks 115 11.2 The Hilbert Spaces L2(R) (or L2(Rd)) and H1(R) (or H1(Rd)) 116 11.3 Dynamical Variables and Hermitian Operators 118 11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum 121 11.5 Observables and Statistical Distributions 125 11.6 The Continuous Spectrum 127 11.7 The Generalized Uncertainty Principle for Dynamical Variables 128 11.7.1 Simultaneous Eigenfunctions 130 12 Applications: The Harmonic Oscillator and the Hydrogen Atom 131 12.1 Introductory Remarks 131 12.2 Ground States and Energy Quanta: The Harmonic Oscillator 131 12.3 The Hydrogen Atom 133 12.3.1 Schrödinger Equation in Spherical Coordinates 135 12.3.2 The Radial Equation 136 12.3.3 The Angular Equation 138 12.3.4 The Orbitals of the Hydrogen Atom 140 12.3.5 Spectroscopic States 140 13 Spin and Pauli's Principle 145 13.1 Angular Momentum and Spin 145 13.2 Extrinsic Angular Momentum 147 13.2.1 The Ladder Property: Raising and Lowering States 149 13.3 Spin 151 13.4 Identical Particles and Pauli's Principle 155 13.5 The Helium Atom 158 13.6 Variational Principle 161 14 Atomic and Molecular Structure 165 14.1 Introduction 165 14.2 Electronic Structure of Atomic Elements 165 14.3 The Periodic Table 169 14.4 Atomic Bonds and Molecules 173 14.5 Examples of Molecular Structures 180 15 Ab Initio Methods: Approximate Methods and Density Functional Theory 189 15.1 Introduction 189 15.2 The Born-Oppenheimer Approximation 190 15.3 The Hartree and the Hartree-Fock Methods 194 15.3.1 The Hartree Method 196 15.3.2 The Hartree-Fock Method 196 15.3.3 The Roothaan Equations 199 15.4 Density Functional Theory 200 15.4.1 Electron Density 200 15.4.2 The Hohenberg-Kohn Theorem 205 15.4.3 The Kohn-Sham Theory 208 III Statistical Mechanics 213 16 Basic Concepts: Ensembles, Distribution Functions, and Averages 215 16.1 Introductory Remarks 215 16.2 Hamiltonian Mechanics 216 16.2.1 The Hamiltonian and the Equations of Motion 218 16.3 Phase Functions and Time Averages 219 16.4 Ensembles, Ensemble Averages, and Ergodic Systems 220 16.5 Statistical Mechanics of Isolated Systems 224 16.6 The Microcanonical Ensemble 228 16.6.1 Composite Systems 230 16.7 The Canonical Ensemble 234 16.8 The Grand Canonical Ensemble 239 16.9 Appendix: A Brief Account of Molecular Dynamics 240 16.9.1 Newtonian's Equations of Motion 241 16.9.2 Potential Functions 242 16.9.3 Numerical Solution of the Dynamical System 245 17 Statistical Mechanics Basis of Classical Thermodynamics 249 17.1 Introductory Remarks 249 17.2 Energy and the First Law of Thermodynamics 250 17.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes 251 17.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics 254 17.4.1 Statistical Interpretation of Q 256 17.5 Entropy and the Partition Function 257 17.6 Conjugate Hamiltonians 259 17.7 The Gibbs Relations 261 17.8 Monte Carlo and Metropolis Methods 262 17.8.1 The Partition Function for a Canonical Ensemble 263 17.8.2 The Metropolis Method 264 17.9 Kinetic Theory: Boltzmann's Equation of Nonequilibrium Statistical Mechanics 265 17.9.1 Boltzmann's Equation 265 17.9.2 Collision Invariants 268 17.9.3 The Continuum Mechanics of Compressible Fluids and Gases: The Macroscopic Balance Laws 269 Exercises 273 Bibliography 317 Index 325