ESSENTIAL PHYSICS: RELATIVITY, PARTICLE DYNAMICS, GRAVITATION, AND WAVE MOTION
CONTENTS 1 MATHEMATICAL PRELIMINARIES 1 1 INVARIANTS 1 2 SOME GEOMETRICAL INVARIANTS 1 3 ELEMENTS OF DIFFERENTIAL GEOMETRY 1 4 GAUSSIAN COORDINATES AND THE INVARIANT LINE ELEMENT 1 5 GEOMETRY AND GROUPS 1 6 VECTORS 1 7 QUATERNIONS 1 8 3-VECTOR ANALYSIS 1 9 LINEAR ALGEBRA AND N-VECTORS 1 10 THE GEOMETRY OF VECTORS 1 11 LINEAR OPERATORS AND MATRICES 1 12 ROTATION OPERATORS 1 13 COMPONENTS OF A VECTOR UNDER COORDINATE ROTATIONS 2 KINEMATICS: THE GEOMETRY OF MOTION 2 1 VELOCITY AND ACCELERATION 2 2 DIFFERENTIAL EQUATIONS OF KINEMATICS 2 3 VELOCITY IN CARTESIAN AND POLAR COORDINATES 2 4 ACCELERATION IN CARTESIAN AND POLAR COORDINATES 3 CLASSICAL AND SPECIAL RELATIVITY 3 1 THE GALILEAN TRANSFORMATION 3 2 EINSTEIN’S SPACE-TIME SYMMETRY: THE LORENTZ TRANSFORMATION 3 3 THE INVARIANT INTERVAL: CONTRAVARIANT AND COVARIANT VECTORS 3 4 THE GROUP STRUCTURE OF LORENTZ TRANSFORMATIONS 3 5 THE ROTATION GROUP 3 6 THE RELATIVITY OF SIMULTANEITY: TIME DILATION AND LENGTH CONTRACTION 3 7 THE 4-VELOCITY 4 NEWTONIAN DYNAMICS 4 1 THE LAW OF INERTIA 4 2 NEWTON’S LAWS OF MOTION 4 3 SYSTEMS OF MANY INTERACTING PARTICLES: CONSERVATION OF LINEAR AND ANGULAR MOMENTUM 4 4 WORK AND ENERGY IN NEWTONIAN DYNAMICS 4 5 POTENTIAL ENERGY 4 6 PARTICLE INTERACTIONS 4 7 THE MOTION OF RIGID BODIES 4 8 ANGULAR VELOCITY AND THE INSTANTANEOUS CENTER OF ROTATION 4 9 AN APPLICATION OF THE NEWTONIAN METHOD 5 INVARIANCE PRINCIPLES AND CONSERVATION LAWS 5 1 INVARIANCE OF THE POTENTIAL UNDER TRANSLATIONS AND THE CONSERVATION OF LINEAR MOMENTUM 5 2 INVARIANCE OF THE POTENTIAL UNDER ROTATIONS AND THE CONSERVATION OF ANGULAR MOMENTUM 6 EINSTEINIAN DYNAMICS 6 1 4-MOMENTUM AND THE ENERGY-MOMENTUM INVARIANT 6 2 THE RELATIVISTIC DOPPLER SHIFT 6 3 RELATIVISTIC COLLISIONS AND THE CONSERVATION OF 4- MOMENTUM 6 4 RELATIVISTIC INELASTIC COLLISIONS 6 5 THE MANDELSTAM VARIABLES 6 6 POSITRON-ELECTRON ANNIHILATION-IN-FLIGHT 7 NEWTONIAN GRAVITATION 7 1 PROPERTIES OF MOTION ALONG CURVED PATHS IN THE PLANE 7 2 AN OVERVIEW OF NEWTONIAN GRAVITATION 7 3 GRAVITATION: AN EXAMPLE OF A CENTRAL FORCE 7 4 MOTION UNDER A CENTRAL FORCE AND THE CONSERVATION OF ANGULAR MOMENTUM 7 5 KEPLER’S 2ND LAW EXPLAINED 7 6 CENTRAL ORBITS 7 7 BOUND AND UNBOUND ORBITS 7 8 THE CONCEPT OF THE GRAVITATIONAL FIELD 7 9 THE GRAVITATIONAL POTENTIAL 8 EINSTEINIAN GRAVITATION: AN INTRODUCTION TO GENERAL RELATIVITY 8 1 THE PRINCIPLE OF EQUIVALENCE 8 2 TIME AND LENGTH CHANGES IN A GRAVITATIONAL FIELD 8 3 THE SCHWARZSCHILD LINE ELEMENT 8 4 THE METRIC IN THE PRESENCE OF MATTER 8 5 THE WEAK FIELD APPROXIMATION 8 6 THE REFRACTIVE INDEX OF SPACE-TIME IN THE PRESENCE OF MASS 8 7 THE DEFLECTION OF LIGHT GRAZING THE SUN 9 AN INTRODUCTION TO THE CALCULUS OF VARIATIONS 9 1 THE EULER EQUATION 9 2 THE LAGRANGE EQUATIONS 9 3 THE HAMILTON EQUATIONS 10 CONSERVATION LAWS, AGAIN 10 1 THE CONSERVATION OF MECHANICAL ENERGY 10 2 THE CONSERVATION OF LINEAR AND ANGULAR MOMENTUM 11 CHAOS 11 1 THE GENERAL MOTION OF A DAMPED, DRIVEN PENDULUM 11 2 THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 12 WAVE MOTION 12 1 THE BASIC FORM OF A WAVE 12 2 THE GENERAL WAVE EQUATION 12 3 THE LORENTZ INVARIANT PHASE OF A WAVE AND THE RELATIVISTIC DOPPLER SHIFT 12 4 PLANE HARMONIC WAVES 12 5 SPHERICAL WAVES 12 6 THE SUPERPOSITION OF HARMONIC WAVES 12 7 STANDING WAVES 13 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 13 1 DEFINITIONS 13 2 SOME TRIGONOMETRIC IDENTITIES AND THEIR FOURIER SERIES 13 3 DETERMINATION OF THE FOURIER COEFFICIENTS OF A FUNCTION 13 4 THE FOURIER SERIES OF A PERIODIC SAW-TOOTH WAVEFORM