Essential Physics 1下载

CONTENTS 1 MATHEMATICAL PRELIMINARIES 1.1 Invariants 1 1.2 Some geometrical invariants 2 1.3 Elements of differential geometry 5 1.4 Gaussian coordinates and the invariant line element 7 1.5 Geometry and groups 10 1.6 Vectors 13 1.7 Quaternions 13 1.8 3-vector analysis 16 1.9 Linear algebra and n-vectors 18 1.10 The geometry of vectors 21 1.11 Linear operators and matrices 24 1.12 Rotation operators 25 1.13 Components of a vector under coordinate rotations 27 2 KINEMATICS: THE GEOMETRY OF MOTION 2.1 Velocity and acceleration 33 2.2 Differential equations of kinematics 36 2.3 Velocity in Cartesian and polar coordinates 39 2.4 Acceleration in Cartesian and polar coordinates 41 3 CLASSICAL AND SPECIAL RELATIVITY 3.1 The Galilean transformation 46 3.2 Einstein’s space-time symmetry: the Lorentz transformation 48 3.3 The invariant interval: contravariant and covariant vectors 51 3.4 The group structure of Lorentz transformations 53 3.5 The rotation group 56 3.6 The relativity of simultaneity: time dilation and length contraction 57 3.7 The 4-velocity 61 4 NEWTONIAN DYNAMICS 4.1 The law of inertia 65 4.2 Newton’s laws of motion 67 4.3 Systems of many interacting particles: conservation of linear and angular vii momentum 68 4.4 Work and energy in Newtonian dynamics 74 4.5 Potential energy 76 4.6 Particle interactions 79 4.7 The motion of rigid bodies 84 4.8 Angular velocity and the instantaneous center of rotation 86 4.9 An application of the Newtonian method 88 5 INVARIANCE PRINCIPLES AND CONSERVATION LAWS 5.1 Invariance of the potential under translations and the conservation of linear momentum 94 5.2 Invariance of the potential under rotations and the conservation of angular momentum 94 6 EINSTEINIAN DYNAMICS 6.1 4-momentum and the energy-momentum invariant 97 6.2 The relativistic Doppler shift 98 6.3 Relativistic collisions and the conservation of 4- momentum 99 6.4 Relativistic inelastic collisions 102 6.5 The Mandelstam variables 103 6.6 Positron-electron annihilation-in-flight 106 7 NEWTONIAN GRAVITATION 7.1 Properties of motion along curved paths in the plane 111 7.2 An overview of Newtonian gravitation 113 7.3 Gravitation: an example of a central force 118 7.4 Motion under a central force and the conservation of angular momentum 120 7.5 Kepler’s 2nd law explained 120 7.6 Central orbits 121 7.7 Bound and unbound orbits 126 7.8 The concept of the gravitational field 128 7.9 The gravitational potential 131 8 EINSTEINIAN GRAVITATION: AN INTRODUCTION TO GENERAL RELATIVITY 8.1 The principle of equivalence 136 8.2 Time and length changes in a gravitational field 138 8.3 The Schwarzschild line element 138 8.4 The metric in the presence of matter 141 8.5 The weak field approximation 142 viii 8.6 The refractive index of space-time in the presence of mass 143 8.7 The deflection of light grazing the sun 144 9 AN INTRODUCTION TO THE CALCULUS OF VARIATIONS 9.1 The Euler equation 149 9.2 The Lagrange equations 151 9.3 The Hamilton equations 153 10 CONSERVATION LAWS, AGAIN 10.1 The conservation of mechanical energy 158 10.2 The conservation of linear and angular momentum 158 11 CHAOS 11.1 The general motion of a damped, driven pendulum 161 11.2 The numerical solution of differential equations 163 12 WAVE MOTION 12.1 The basic form of a wave 167 12.2 The general wave equation 170 12.3 The Lorentz invariant phase of a wave and the relativistic Doppler shift 171 12.4 Plane harmonic waves 173 12.5 Spherical waves 174 12.6 The superposition of harmonic waves 176 12.7 Standing waves 177 13 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 13.1 Definitions 179 13.2 Some trigonometric identities and their Fourier series 180 13.3 Determination of the Fourier coefficients of a function 182 13.4 The Fourier series of a periodic saw-tooth waveform 183 APPENDIX A SOLVING ORDINARY DIFFERENTIAL EQUATIONS 187 BIBLIOGRAPHY 198