ESSENTIAL PHYSICS Part 1 - FRANK W.K.FIRK.pdf

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ESSENTIAL PHYSICS
Part 1
R EL AT IV IT Y, P AR TI CL E DY NA MI CS , G R AV IT AT IO N,
A ND W AV E MO TI ON
F R A N K W . K . F I R K
P r o f e s s o r E m e r i t u s o f P h y s i c s
Y a l e U n i v e r s i t y
2000
ii
iii
CONTENTS
PREFACE
vii
1 MATHEMATICAL PRELIMINARIES
1.2 Some geometrical invariants
1
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
2.2 Differential equations of kinematics
31
2.3 Velocity in Cartesian and polar coordinates
37
2.4 Acceleration in Cartesian and polar coordinates
39
3 CLASSICAL AND SPECIAL RELATIVITY
3.2 Einstein’s space-time symmetry: the Lorentz transformation
44
3.3 The invariant interval: contravariant and covariant vectors
49
3.4 The group structure of Lorentz transformations
51
3.5 The rotation group
54
3.6 The relativity of simultaneity: time dilation and length contraction
55
3.7 The 4-velocity
59
4 NEWTONIAN DYNAMICS
4.1 The law of inertia
63
1.1 Invariants
2
2.1 Velocity and acceleration
34
3.1 The Galilean transformation
46
iv
4.3 Systems of many interacting particles: conservation of linear and angular
momentum
65
4.4 Work and energy in Newtonian dynamics
66
4.5 Potential energy
74
4.6 Particle interactions
77
4.7 The motion of rigid bodies
81
4.8 Angular velocity and the instantaneous center of rotation
84
4.9 An application of the Newtonian method
86
5 INVARIANCE PRINCIPLES AND CONSERVATION LAWS
5.1 Invariance of the potential under translations and the conservation of linear
momentum
93
93
6 EINSTEINIAN DYNAMICS
6.1 4-momentum and the energy-momentum invariant
96
6.3 Relativistic collisions and the conservation of 4- momentum
98
6.4 Relativistic inelastic collisions
101
6.5 The Mandelstam variables
102
6.6 Positron-electron annihilation-in-flight
105
7 NEWTONIAN GRAVITATION
7.1 Properties of motion along curved paths in the plane 110
7.2 An overview of Newtonian gravitation 112
7.3 Gravitation: an example of a central force 117
7.4 Motion under a central force and the conservation of angular momentum 119
7.5 Kepler’s 2nd law explained
119
7.7 Bound and unbound orbits
125
7.8 The concept of the gravitational field
126
7.9 The gravitational potential
130
8 EINSTEINIAN GRAVITATION: AN INTRODUCTION TO GENERAL RELATIVITY
8.2 Time and length changes in a gravitational field
135
8.3 The Schwarzschild line element
137
4.2 Newton’s laws of motion
72
5.2 Invariance of the potential under rotations and the conservation of angular
momentum
6.2 The relativistic Doppler shift
97
7.6 Central orbits
120
8.1 The principle of equivalence
137
v
8.4 The metric in the presence of matter
140
8.6 The refractive index of space-time in the presence of mass
142
8.7 The deflection of light grazing the sun
143
9 AN INTRODUCTION TO THE CALCULUS OF VARIATIONS
9.2 The Lagrange equations
148
9.3 The Hamilton equations
152
10 CONSERVATION LAWS, AGAIN
10.2 The conservation of linear and angular momentum
156
11 CHAOS
11.1 The general motion of a damped, driven pendulum
160
12 WAVE MOTION
12.2 The general wave equation
166
12.3 The Lorentz invariant phase of a wave and the relativistic Doppler shift
170
12.4 Plane harmonic waves
172
12.5 Spherical waves
173
12.6 The superposition of harmonic waves
175
12.7 Standing waves
176
13 ORTHOGONAL FUNCTIONS AND FOURIER SERIES
13.2 Some trigonometric identities and their Fourier series
178
13.3 Determination of the Fourier coefficients of a function
181
13.4 The Fourier series of a periodic saw-tooth waveform
182
APPENDIX A SOLVING ORDINARY DIFFERENTIAL EQUATIONS
186
BIBLIOGRAPHY
197
8.5 The weak field approximation
141
9.1 The Euler equation
150
10.1 The conservation of mechanical energy
157
11.2 The numerical solution of differential equations
162
12.1 The basic form of a wave
169
13.1 Definitions
179

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