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.

PrefaceChapter 1 Gravity 1.1 Introduction 1.2 GravityChapter 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 RemarkChapter 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 RemarkChapter 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 FlowsChapter 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 FlowChapter 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 TransformationChapter 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 RemarksChapter 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 DynamicsChapter 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 FieldAppendix 1 Fortran Program for General N-Body InteractionAppendix 2 Fortran Program for Planetary-Type EvolutionReferences and Sources for Further ReadingIndex