Arithmetic Applied Mathematics (eBook)

Donald Greenspan (Autor)

V. Lakshmikantham, C P Tsokos (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
174 Seiten
Elsevier Science (Verlag)
978-1-4832-7985-5 (ISBN)

Infos

Arithmetic Applied Mathematics deals with concepts of arithmetic applied mathematics and uses a computer, rather than a continuum, approach to the deterministic theories of particle mechanics. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics using only arithmetic. Definitions of energy and momentum are presented that are identical to those of continuum mechanics. Comprised of nine chapters, this book begins by exploring discrete modeling as it relates to Newtonian mechanics and special relativistic mechanics, paying particular attention to gravity. The reader is then introduced to long-range forces such as gravitation and short-range forces such as molecular attraction and repulsion; the N-body problem; and conservative and non-conservative models of complex physical phenomena. Subsequent chapters focus on the foundational concepts of special relativity; arithmetic special relativistic mechanics in one space dimension and three space dimensions; and Lorentz invariant computations. This monograph will be of interest to students and practitioners in the fields of mathematics and physics.

Inhalt

Front Cover 1
Arithmetic Applied Mathematics 4
Copyright Page 
5 
Table of Contents 6
Preface 8
Chapter 1. Gravity 10
1.1 INTRODUCTION 10
1.2 GRAVITY 10
Chapter 2. Long and Short Range Forces: Gravitation and Molecular Attraction and Repulsion 16
2.1 INTRODUCTION 16
2.2 GRAVITATION 16
2.3 BASIC PLANAR CONCEPTS 18
2.4 DISCRETE GRAVITATION AND PLANETARY MOTION 20
2.5 THE GENERALIZED NEWTON'S METHOD 22
2.6 AN ORBIT EXAMPLE 23
2.7 GRAVITY REVISITED 25
2.8 CLASSICAL MOLECULAR FORCES 27
2.9 REMARK 28
Chapter 3. The N-body Problem 29
3.1 INTRODUCTION 29
3.2 THE THREE-BODY PROBLEM 29
3.3 CONSERVATION OF ENERGY 31
3.4 SOLUTION OF THE DISCRETE THREE-BODY PROBLEM 32
3.5 CENTER OF GRAVITY 34
3.6 CONSERVATION OF LINEAR MOMENTUM 36
3.7 CONSERVATION OF ANGULAR MOMENTUM 36
3.8 THE N-BODY PROBLEM 39
3.9 REMARK 40
Chapter 4. Conservative Models 41
4.1 INTRODUCTION 41
4.2 THE SOLID STATE BUILDING BLOCK 41
4.3 FLOW OF HEAT IN A BAR 43
4.4 OSCILLATION OF AN ELASTIC BAR 45
4.5 LAMINAR AND TURBULENT FLUID FLOWS 48
Chapter 5. Nonconservative Models 53
5.1 INTRODUCTION 53
5.2 SHOCK WAVES 53
5.3 THE LEAP-FROG FORMULAS 57
5.4 THE STEFAN PROBLEM 57
5.5 EVOLUTION OF PLANETARY TYPE BODIES 73
5.6 FREE SURFACE FLUID FLOW 101
5.7 POROUS FLOW 113
Chapter 6. Foundational Concepts of Special Relativity 118
6.1 INTRODUCTION 118
6.2 BASIC CONCEPTS 118
6.3 EVENTS AND A SPECIAL LORENTZ TRANSFORMATION 119
6.4 A GENERAL LORENTZ TRANSFORMATION 121
Chapter 7. Arithmetic Special Relativistic Mechanics in One Space Dimension 124
7.1 INTRODUCTION 124
7.2 PROPER TIME 125
7.3 VELOCITY AND ACCELERATION 126
7.4 REST MASS AND MOMENTUM 128
7.5 THE DYNAMICAL DIFFERENCE EQUATION 129
7.6 ENERGY 130
7.7 THE MOMENTUM-ENERGY VECTOR 132
7.8 REMARKS 133
Chapter 8. Arithmetic Special Relativistic Mechanics in Three Space Dimensions 134
8.1 INTRODUCTION 134
8.2 VELOCITY, ACCELERATION, AND PROPER TIME 134
8.3 MINKOWSKI SPACE 137
8.4 4-VELOCITY AND 4-ACCELERATION 138
8.5 MOMENTUM AND ENERGY 140
8.6 THE MOMENTUM-ENERGY 4-VECTOR 141
8.7 DYNAMICS 142
Chapter 9. Lorentz Invariant Computations 145
9.1 INTRODUCTION 145
9.2 INVARIANT COMPUTATIONS 145
9.3 AN ARITHMETIC, NEWTONIAN HARMONIC OSCILLATOR 147
9.4 AN ARITHMETIC, RELATIVISTIC HARMONIC OSCILLATOR 150
9.5 MOTION OF AN ELECTRIC CHARGE IN A MAGNETIC FIELD 154
Appendix 1: Fortran Program for General N-body Interaction 157
Appendix 2: Fortran Program for Planetary-type Evolution 163
References and Sources for Further Reading 167
Index 172

Produkt-Details

• Erscheint lt. Verlag: 9.5.2014
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• Themenwelt: Technik
• ISBN-10: 1-4832-7985-5 / 1483279855
• ISBN-13: 978-1-4832-7985-5 / 9781483279855