N-BODY PROBLEMS AND MODELS

By JOSEPH JOHNSON

CONTENTS 1 THE 1-BODY PROBLEM 1 1 NONLINEAR OSCILLATION 1 2 CONCEPTS FROM SPECIAL RELATIVITY 1 3 RELATIVISTIC OSCILLATION 1 4 NUMERICAL METHODOLOGY 1 5 REMARK 2 1 INTRODUCTION 2 2 NUMERICAL METHODOLOGY 2 3 CONSERVATION LAWS 2 4 COVARIANCE 2 5 PERIHELION MOTION 2 6 THE FUNDAMENTAL PROBLEM OF ELECTROSTATICS 2 7 THE CALOGERO HAMILTONIAN SYSTEM 2 8 THE TODA HAMILTONIAN SYSTEM 2 9 REMARKS 3 1 CLASSICAL MOLECULAR FORCES 3 2 EQUATIONS OF MOTION FOR WATER VAPOR MOLECULES 3 3 A CAVITY PROBLEM 3 4 COMPUTATIONAL CONSIDERATIONS 3 5 PRIMARY VORTEX GENERATION 3 6 TURBULENT FLOW GENERATION 3 7 PRIMARY VORTICES AND TURBULENCE FOR AIR 3 8 SIMULATION OF CRACKS AND FRACTURES IN A SHEET OF ICE 3 9 SHOCKS IN A NANOTUBE 4 N(NUMBER OF MOLECULES) > 10000 THE CAVITY PROBLEM 4 1 INTRODUCTION 4 2 PARTICLE ARRANGEMENT AND EQUATIONS FOR WATER VAPOR 4 3 PARTICLE EQUILIBRIUM 4 4 EXAMPLES 5 N(NUMBER OF MOLECULES) > 10000 CRACK AND FRACTURE DEVELOPMENT 5 1 INTRODUCTION 5 2 FORMULA DERIVATION 5 3 EXAMPLE 6 N(MOLECULES) > 10000 CONTACT ANGLE OF ADHESION 6 1 INTRODUCTION 6 2 LOCAL FORCE FORMULAS 6 3 DYNAMICAL EQUATIONS 6 4 DROP AND SLAB STABILIZATION 6 5 SESSILE DROP FORMATION 6 6 REMARK 7 A PARTICLE MODEL OF CARBON DIOXIDE BUBBLES IN WATER 7 1 INTRODUCTION 7 2 FORMULATION OF THE PARTICLE MODEL 7 3 BASIN OF WATER STABILIZATION 7 4 CARBON DIOXIDE BUBBLES AND THEIR MOTIONS 8 A PARTICLE MODEL A DODECAHEDRAL ROTATING TOP 8 1 INTRODUCTION 8 2 A DISCRETE, DODECAHEDRAL TOP 8 3 DYNAMICAL EQUATIONS 8 4 NUMERICAL METHOD 8 5 EXAMPLES 8 6 REMARKS 9 A PARTICLE MODEL OF SELF REORGANIZATION 9 1 INTRODUCTION 9 2 THE COMPUTER ALGORITHM 9 3 EXAMPLE 10 PARTICLE MODEL OF A BOUNCING ELASTIC BALL 10 1 INTRODUCTION 10 2 GENERATING A BALL AS A PARTICLE MODEL 10 3 EXAMPLES 11 PARTICLE MODEL OF STRING SOLITONS 11 1 INTRODUCTION 11 2 DISCRETE STRINGS 11 3 EXAMPLES 11 4 REMARK 12 PARTICLE MODELS OF MINIMAL SURFACES AND SADDLE SURFACES 12 1 INTRODUCTION 12 2 A PARTICLE MODEL OF A MINIMAL SURFACE 12 3 A MONKEY SADDLE