# Arithmetic Applied Mathematics

## 1st Edition

### International Series in Nonlinear Mathematics: Theory, Methods and Applications

**Authors:**Donald Greenspan

**Editors:**V. Lakshmikantham C P Tsokos

**eBook ISBN:**9781483138305

**Imprint:**Pergamon

**Published Date:**1st January 1980

**Page Count:**174

## Description

Arithmetic Applied Mathematics deals with the deterministic theories of particle mechanics using a computer approach. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics with the aid only of arithmetic. The computational power of modern digital computers is highlighted, along with simple models of complex physical phenomena and solvable dynamical equations for both linear and nonlinear behavior.
This book is comprised of nine chapters and opens by describing an experiment with gravity, followed by a discussion on the two basic types of forces that are important in classical physical modeling: long range forces and short range forces. Gravitation and molecular attraction and repulsion are considered, along with the basic concepts of position, velocity, and acceleration. The reader is then introduced to the N-body problem; conservative and non-conservative models of complex physical phenomena; foundational concepts of special relativity; and arithmetic special relativistic mechanics in one space dimension and three space dimensions. The final chapter is devoted to Lorentz invariant computations, with emphasis on the arithmetic modeling and analysis of a harmonic oscillator.

This monograph will be of interest to mathematicians, physicists, and computer scientists.

## Table of Contents

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 Relativity

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 Mechanics 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

## Details

- No. of pages:
- 174

- Language:
- English

- Copyright:
- © Pergamon 1980

- Published:
- 1st January 1980

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483138305

## About the Author

### Donald Greenspan

## About the Editor

### V. Lakshmikantham

### Affiliations and Expertise

University of Texas at Arlington, USA