# An Introduction to Mathematical Modeling. A Course in Mechanics. Wiley Series in Computational Mechanics

- ID: 2176674
- November 2011
- 348 Pages
- John Wiley and Sons Ltd

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
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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(M) (or L2(Rd)) and Í\Ø) (or H\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

The book also serves as a valuable reference for professionals working in the areas of modeling and simulation, physics, and computational engineering. (Zentralblatt MATH, 2012)