An Introduction to Mathematical Modeling E-Book

J. Tinsley Oden

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Computational Mechanics
• Sprache: Englisch

Seitenzahl: 285

Veröffentlichungsjahr: 2012

Beschreibung

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.

Leseprobe

CONTENTS

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Part I: Nonlinear Continuum Mechanics

Chapter 1: Kinematics of Deformable Bodies

1.1 Motion

1.2 Strain and Deformation Tensors

1.3 Rates of Motion

1.4 Rates of Deformation

1.5 The Piola Transformation

1.6 The Polar Decomposition Theorem

1.7 Principal Directions and Invariants of Deformation and Strain

1.8 The Reynolds’ Transport Theorem

Chapter 2: Mass and Momentum

2.1 Local Forms of the Principle of Conservation of Mass

2.2 Momentum

Chapter 3: Force and Stress in Deformable Bodies

Chapter 4: The Principles of Balance of Linear and Angular Momentum

4.1 Cauchy’s Theorem: The Cauchy Stress Tensor

4.2 The Equations of Motion (Linear Momentum)

4.3 The Equations of Motion Referred to the Reference Configuration: The Piola–Kirchhoff Stress Tensors

4.4 Power

Chapter 5: The Principle of Conservation of Energy

5.1 Energy and the Conservation of Energy

5.2 Local Forms of the Principle of Conservation of Energy

Chapter 6: Thermodynamics of Continua and the Second Law

Chapter 7: Constitutive Equations

7.1 Rules and Principles for Constitutive Equations

7.2 Principle of Material Frame Indifference

7.3 The Coleman–Noll Method: Consistency with the Second Law of Thermodynamics

Chapter 8: Examples and Applications

8.1 The Navier–Stokes Equations for Incompressible Flow

8.2 Flow of Gases and Compressible Fluids: The Compressible Navier–Stokes Equations

8.3 Heat Conduction

8.4 Theory of Elasticity

Part II: Electromagnetic Field Theory and Quantum Mechanics

Chapter 9: Electromagnetic Waves

9.1 Introduction

9.2 Electric Fields

9.3 Gauss’s Law

9.4 Electric Potential Energy

9.5 Magnetic Fields

9.6 Some Properties of Waves

9.7 Maxwell’s Equations

9.8 Electromagnetic Waves

Chapter 10: Introduction to Quantum Mechanics

10.1 Introductory Comments

10.2 Wave and Particle Mechanics

10.3 Heisenberg’s Uncertainty Principle

10.4 Schrödinger’s Equation

10.5 Elementary Properties of the Wave Equation

10.6 The Wave–Momentum Duality

10.7 Appendix: A Brief Review of Probability Densities

Chapter 11: Dynamical Variables and Observables in Quantum Mechanics: The Mathematical Formalism

11.1 Introductory Remarks

11.2 The Hilbert Spaces L2() (or L2(d)) and H1() (or H1(d))

11.3 Dynamical Variables and Hermitian Operators

11.4 Spectral Theory of Hermitian Operators: The Discrete Spectrum

11.5 Observables and Statistical Distributions

11.6 The Continuous Spectrum

11.7 The Generalized Uncertainty Principle for Dynamical Variables

Chapter 12: Applications: The Harmonic Oscillator and the Hydrogen Atom

12.1 Introductory Remarks

12.2 Ground States and Energy Quanta: The Harmonic Oscillator

12.3 The Hydrogen Atom

Chapter 13: Spin and Pauli’s Principle

13.1 Angular Momentum and Spin

13.2 Extrinsic Angular Momentum

13.3 Spin

13.4 Identical Particles and Pauli’s Principle

13.5 The Helium Atom

13.6 Variational Principle

Chapter 14: Atomic and Molecular Structure

14.1 Introduction

14.2 Electronic Structure of Atomic Elements

14.3 The Periodic Table

14.4 Atomic Bonds and Molecules

14.5 Examples of Molecular Structures

Chapter 15: Ab Initio Methods: Approximate Methods and Density Functional Theory

15.1 Introduction

15.2 The Born–Oppenheimer Approximation

15.3 The Hartree and the Hartree–Fock Methods

15.4 Density Functional Theory

Part III: Statistical Mechanics

Chapter 16: Basic Concepts: Ensembles, Distribution Functions, and Averages

16.1 Introductory Remarks

16.2 Hamiltonian Mechanics

16.3 Phase Functions and Time Averages

16.4 Ensembles, Ensemble Averages, and Ergodic Systems

16.5 Statistical Mechanics of Isolated Systems

16.6 The Microcanonical Ensemble

16.7 The Canonical Ensemble

16.8 The Grand Canonical Ensemble

16.9 Appendix: A Brief Account of Molecular Dynamics

Chapter 17: Statistical Mechanics Basis of Classical Thermodynamics

17.1 Introductory Remarks

17.2 Energy and the First Law of Thermodynamics

17.3 Statistical Mechanics Interpretation of the Rate of Work in Quasi-Static Processes

17.4 Statistical Mechanics Interpretation of the First Law of Thermodynamics

17.5 Entropy and the Partition Function

17.6 Conjugate Hamiltonians

17.7 The Gibbs Relations

17.8 Monte Carlo and Metropolis Methods

17.9 Kinetic Theory: Boltzmann’s Equation of Nonequilibrium Statistical Mechanics

Exercises

Exercises for Part I

Exercise Set I.1

Exercise Set I.2

Exercise Set I.3

Exercise Set I.4

Exercise Set I.5

Exercise Set I.6

Exercises for Part II

Exercise Set II.1

Exercise Set II.2

Exercise Set II.3

Exercise Set II.4

Exercises for Part III

Exercise Set III.1

Exercise Set III.2

Bibliography

Part I

Part II

Part III

Index

An Introduction to Mathematical Modeling

WILEY SERIES IN COMPUTATIONAL MECHANICS

Series Advisors: René de Borst, Perumal Nithiarasu, Tayfun Tezduyar, Genki Yagawa, Tarek Zohdi

Introduction to Finite Element Analysis: Formulation, Verification and ValidationBarna Szabó, Ivo Babuška (March 2011)

An Introduction to Mathematical Modeling: A Course in MechanicsJ. Tinsley Oden (September 2011)

Computational Mechanics of DiscontinuaAntonio A Munjiza, Earl Knight, Esteban Rougier (October 2011)

Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Oden, J. Tinsley (John Tinsley), 1936–An introduction to mathematical modeling: a course in mechanics / J. Tinsley Oden.p. cm. — (Wiley series in computational mechanics)Includes bibliographical references and index.ISBN 978-1-118-01903-0 (hardback)1. Mechanics, Analytic. I. Title.QA807.O34 2011530—dc23                            2011012204

oBook: 978-1-118-10573-3ePDF: 978-1-118-10576-4ePub: 978-1-118-10574-0

To Walker and Lee

PREFACE

This text was written for a course on An Introduction to Mathematical Modeling for students with diverse backgrounds in science, mathematics, and engineering who enter our program in Computational Science, Engineering, and Mathematics. It is not, however, a course on just how to construct mathematical models of physical phenomena. It is a course designed to survey the classical mathematical models of subjects forming the foundations of modern science and engineering at a level accessible to students finishing undergraduate degrees or entering graduate programs in computational science. Along the way, I develop through examples how the most successful models in use today arise from basic principles and modern and classical mathematics. Students are expected to be equipped with some knowledge of linear algebra, matrix theory, vector calculus, and introductory partial differential equations, but those without all these prerequisites should be able to fill in some of the gaps by doing the exercises.

I have chosen to call this a textbook on mechanics, since it covers introductions to continuum mechanics, electrodynamics, quantum mechanics, and statistical mechanics. If mechanics is the branch of physics and mathematical science concerned with describing the motion of bodies, including their deformation and temperature changes, under the action of forces, and if one adds to this the study of the propagation of waves and the transformation of energy in physical systems, then the term mechanics does indeed apply to everything that is covered here.

The course is divided into three parts. Part I is a short course on nonlinear continuum mechanics; Part II contains a brief account of electromagnetic wave theory and Maxwell’s equations, along with an introductory account of quantum mechanics, pitched at an undergraduate level but aimed at students with a bit more mathematical sophistication than many undergraduates in physics or engineering; and Part III is a brief introduction to statistical mechanics of systems, primarily those in thermodynamic equilibrium.

There are many good treatments of the component parts of this work that have contributed to my understanding of these subjects and inspired their treatment here. The books of Gurtin, Ciarlet, and Batra provide excellent accounts of continuum mechanics at an accessible level, and the excellent book of Griffiths on introductory quantum mechanics is a well-crafted text on this subject. The accounts of statistical mechanics laid out in the book of Weiner and the text of McQuarrie, among others, provide good introductions to this subject. I hope that the short excursion into these subjects contained in this book will inspire students to want to learn more about these subjects and will equip them with the tools needed to pursue deeper studies covered in more advanced texts, including some listed in the references.

The evolution of these notes over a period of years benefited from input from several colleagues. I am grateful to Serge Prudhomme, who proofread early versions and made useful suggestions for improvement. I thank Alex Demkov for reading and commenting on Part II. My sincere thanks also go to Albert Romkes, who helped with early drafts, to Ludovic Chamoin, who helped compile and type early drafts of the material on quantum mechanics, and Kris van der Zee, who helped compile a draft of the manuscript and devoted much time to proofreading and helping with exercises. I am also indebted to Pablo Seleson, who made many suggestions that improved presentations in Part II and Part III and who was of invaluable help in putting the final draft together.

J. Tinsley Oden

Austin, TexasJune 2011

Part I

Nonlinear Continuum Mechanics