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Preface

Chapter 1 Gravity

1.1 Introduction

1.2 Gravity

Chapter 2 Long and Short Range Forces: Gravitation and Molecular Attraction and Repulsion

2.1 Introduction

2.2 Gravitation

2.3 Basic Planar Concepts

2.4 Discrete Gravitation and Planetary Motion

2.5 The Generalized Newton's Method

2.6 An Orbit Example

2.7 Gravity Revisited

2.8 Classical Molecular Forces

2.9 Remark

Chapter 3 The N-Body Problem

3.1 Introduction

3.2 The Three-Body Problem

3.3 Conservation of Energy

3.4 Solution of the Discrete Three-Body Problem

3.5 Center of Gravity

3.6 Conservation of Linear Momentum

3.7 Conservation of Angular Momentum

3.8 The N-Body Problem

3.9 Remark

Chapter 4 Conservative Models

4.1 Introduction

4.2 The Solid State Building Block

4.3 Flow of Heat in a Bar

4.4 Oscillation of an Elastic Bar

4.5 Laminar and Turbulent Fluid Flows

Chapter 5 Nonconservative Models

5.1 Introduction

5.2 Shock Waves

5.3 The Leap-Frog Formulas

5.4 The Stefan Problem

5.5 Evolution of Planetary Type Bodies

5.6 Free Surface Fluid Flow

5.7 Porous Flow

Chapter 6 Foundational Concepts of Special Relativiy

6.1 Introduction

6.2 Basic Concepts

6.3 Events and a Special Lorentz Transformation

6.4 A General Lorentz Transformation

Chapter 7 Arithmetic Special Relativistic Mechanisms in One Space Dimension

7.1 Introduction

7.2 Proper Time

7.3 Velocity and Acceleration

7.4 Rest Mass and Momentum

7.5 The Dynamical Difference Equation

7.6 Energy

7.7 The Momentum-Energy Vector

7.8 Remarks

Chapter 8 Arithmetic Special Relativistic Mechanics in Three Space Dimensions

8.1 Introduction

8.2 Velocity, Acceleration, and Proper Time

8.3 Minkowski Space

8.4 4-Velocity and 4-Acceleration

8.5 Momentum and Energy

8.6 The Momentum-Energy 4-Vector

8.7 Dynamics

Chapter 9 Lorentz Invariant Computations

9.1 Introduction

9.2 Invariant Computations

9.3 An Arithmetic, Newtonian Harmonic Oscillator

9.4 An Arithmetic, Relativistic Harmonic Oscillator

9.5 Motion of an Electric Charge in a Magnetic Field

Appendix 1 Fortran Program for General N-Body Interaction

Appendix 2 Fortran Program for Planetary-Type Evolution

References and Sources for Further Reading

Index

### V. Lakshmikantham

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1st Edition - January 1, 1980

Author: Donald Greenspan

Editors: V. Lakshmikantham, C P Tsokos

Language: EnglisheBook ISBN:

9 7 8 - 1 - 4 8 3 2 - 7 9 8 5 - 5

Arithmetic Applied Mathematics deals with concepts of arithmetic applied mathematics and uses a computer, rather than a continuum, approach to the deterministic theories of… Read more

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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.

Preface

Chapter 1 Gravity

1.1 Introduction

1.2 Gravity

Chapter 2 Long and Short Range Forces: Gravitation and Molecular Attraction and Repulsion

2.1 Introduction

2.2 Gravitation

2.3 Basic Planar Concepts

2.4 Discrete Gravitation and Planetary Motion

2.5 The Generalized Newton's Method

2.6 An Orbit Example

2.7 Gravity Revisited

2.8 Classical Molecular Forces

2.9 Remark

Chapter 3 The N-Body Problem

3.1 Introduction

3.2 The Three-Body Problem

3.3 Conservation of Energy

3.4 Solution of the Discrete Three-Body Problem

3.5 Center of Gravity

3.6 Conservation of Linear Momentum

3.7 Conservation of Angular Momentum

3.8 The N-Body Problem

3.9 Remark

Chapter 4 Conservative Models

4.1 Introduction

4.2 The Solid State Building Block

4.3 Flow of Heat in a Bar

4.4 Oscillation of an Elastic Bar

4.5 Laminar and Turbulent Fluid Flows

Chapter 5 Nonconservative Models

5.1 Introduction

5.2 Shock Waves

5.3 The Leap-Frog Formulas

5.4 The Stefan Problem

5.5 Evolution of Planetary Type Bodies

5.6 Free Surface Fluid Flow

5.7 Porous Flow

Chapter 6 Foundational Concepts of Special Relativiy

6.1 Introduction

6.2 Basic Concepts

6.3 Events and a Special Lorentz Transformation

6.4 A General Lorentz Transformation

Chapter 7 Arithmetic Special Relativistic Mechanisms in One Space Dimension

7.1 Introduction

7.2 Proper Time

7.3 Velocity and Acceleration

7.4 Rest Mass and Momentum

7.5 The Dynamical Difference Equation

7.6 Energy

7.7 The Momentum-Energy Vector

7.8 Remarks

Chapter 8 Arithmetic Special Relativistic Mechanics in Three Space Dimensions

8.1 Introduction

8.2 Velocity, Acceleration, and Proper Time

8.3 Minkowski Space

8.4 4-Velocity and 4-Acceleration

8.5 Momentum and Energy

8.6 The Momentum-Energy 4-Vector

8.7 Dynamics

Chapter 9 Lorentz Invariant Computations

9.1 Introduction

9.2 Invariant Computations

9.3 An Arithmetic, Newtonian Harmonic Oscillator

9.4 An Arithmetic, Relativistic Harmonic Oscillator

9.5 Motion of an Electric Charge in a Magnetic Field

Appendix 1 Fortran Program for General N-Body Interaction

Appendix 2 Fortran Program for Planetary-Type Evolution

References and Sources for Further Reading

Index

- No. of pages: 174
- Language: English
- Edition: 1
- Published: January 1, 1980
- Imprint: Pergamon
- eBook ISBN: 9781483279855

VL

Affiliations and expertise

University of Texas at Arlington, USARead *Arithmetic Applied Mathematics* on ScienceDirect