Dynamics and Relativity

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ISBN: 9780198501121 | 0198501129
Cover: Paperback
Copyright: 2/10/2000

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Special relativity suffers from the myth that it is difficult. In order to overcome this barrier Dynamics and relativity presents an integrated treatment of classical mechanics and special relativity, by treating classical mechanics as Galilean relativity. This gives students the freedom toformulate a particular problem in one frame of reference and solve it in another, where it takes a simpler form. This strategy, which is central to special relativity, is applied to problems in classical mechanics, thus preparing the tools and thought patterns for a treatment of specialrelativity.

Galilean relativity

(30)

Brief reminder of Newton's laws

(3)

Newton's second law as a definition of the concepts of force and mass

(1)

Newton's second law as an equation of motion

(1)

The nature of force

(4)

Conservative forces

(1)

The fundamental forces

(2)

Forces encountered in applications

(1)

Force of gravity

(7)

Newtonian field equations for gravity

(2)

Self-gravitating fluids

(1)

Example: Radial variation of pressure and density in model stars

(2)

The nature of mass

(3)

Inertial mass

(1)

Gravitational mass

(1)

The principle of equivalence

(1)

The space-time continuum

(3)

Measurement of length and time

(1)

Ideal measuring rods and ideal clocks

(1)

Relativity of motion in Newtonian mechanics

(2)

Inertial frames of reference

(1)

Inertial frames in standard configuration

(1)

Galilean transformation

(1)

Galilean invariance of Newton's laws

(1)

The principle of Galilean relativity

(4)

Relativity of speed

(1)

Example: Motion of a projectile relative to S and S'

(1)

Example: Motion of a particle sliding down a slope in S' relative to S

(1)

The instantaneous rest frame of a moving body

(1)

Example: A particle undergoing constant linear acceleration

(1)

Exercises

(2)

Conservation laws

(18)

Conservation of linear momentum

(1)

Conservation of energy

(6)

Graphical treatment of potentials

(1)

Example:Find the potentials for specified force fields

(1)

Example: Time taken for a particle to move a given distance under the influence of a potential field

(1)

Example: Elastic collision of two particles

(1)

Example: Simple pendulum oscillating with finite amplitude

(1)

Angular momentum

(3)

Analogous form of N2 for angular motion

(1)

N1 for angular motion: conservation of angular momentum

(1)

Angular momentum of a particle relative to S and S'

(1)

Example: Angular momentum of a particle with impact parameter b incident from infinity on a centre of force in S and S'

(1)

Two-particle systems

(6)

Momentum

(1)

Centre of mass (CM) and relative coordinates

(1)

Kinetic and potential energy

(2)

The reduced mass and equivalent single-body problem

(1)

Example: Conservation of energy of two particles in terms of the reduced mass

(1)

Verification of the Galilean invariance of N3

(1)

Exercises

(2)

Central forces

(28)

Definition of a central force field

(1)

The two-dimensional problem

(3)

Constant angular momentum

(1)

Motion in a plane

(1)

Velocity of a particle in plane polar coordinates

(1)

Angular momentum in plane polar coordinates

(1)

Conservation of energy for motion under a central force

(1)

Conservation of angular momentum and the equivalent one-dimensional problem

(3)

The effective potential

(1)

Classification of orbits

(1)

Example: An attractive force vanishing at infinity

(1)

Equation for the orbit from conservation of energy

(6)

The inverse-square law: orbits as conic sections

(2)

Example: α-particle scattering

(2)

The inverse problem: obtain the force given the orbit

(1)

Examples: Obtain the forces given the orbits

(1)

Equation of the orbit using N2

(6)

Acceleration in polar coordinates

(1)

The equations of radial and circumferential motion

(1)

Example: Verify that an elliptic orbit corresponds to an inverse-square law of force

(1)

The orbit equation

(1)

Example: Given the orbit, obtain the force

(1)

Example: Given the law of force, find the orbit

(2)

Kepler's laws of planetary motion

(4)

Kepler's third law

(1)

The two-body problem and the Galilean-relativistic corrections to Kepler's third law

(2)

Exercises

(2)

Mechanical vibrations and waves

(26)

Stable and unstable equilibrium

(2)

Small oscillations

(1)

Systems with one degree of freedom

(7)

The simple harmonic oscillator

(1)

The simple pendulum

(1)

The spring--mass system

(1)

A mass on a stretched string

(1)

The effect of damping

(3)

Forced vibrations and resonance

(1)

Systems with several degrees of freedom

(5)

Two masses on a stretched string

(1)

Normal modes of vibration

(1)

Example: The double pendulum

(2)

Waves

(10)

SHM in a moving reference frame

(1)

The mathematical representation of a travelling wave

(2)

Wave speeds in elastic media as obtained by Galilean transformation

(2)

The general form of the wave equation and its solutions

(2)

Travelling waves and standing waves

(1)

Acoustics in moving reference frames: the classical Doppler effect

(2)

Exercises

101

(2)

Systems of N particles

103

(14)

The centre of mass of a system

103

(2)

Example: Centre of mass of three given masses at specified positions

104

(1)

Linear momentum of a system

105

(2)

Example: Velocity of the centre of mass of three particles

106

(1)

Energy of a system

107

(4)

Kinetic energy

108

(1)

Potential energy

108

(1)

Example: Energy of N particles in CM and relative coordinates

108

(2)

Example: Kinetic energy of two particles in terms of the reduced mass

110

(1)

The angular momentum of a system

111

(2)

Torque

113

(2)

Form of N2 for angular motion

113

(2)

Continuous distributions of mass

115

(1)

Example: Centre of mass of a solid hemisphere

115

(1)

Exercises

116

(1)

Solid-body motion

117

(18)

Motion about a fixed axis

117

(1)

Moments of inertia

118

(10)

The moment of inertia / of a body

119

(1)

The radius of gyration of a body

120

(1)

Example: Moment of inertia of a slender rod about an axis perpendicular to its length

121

(1)

Example: Moment of inertia of a solid cylinder about its axis of symmetry

122

(1)

Parallel axes theorem

122

(1)

Perpendicular axes theorem

123

(2)

Example: Radius of gyration of a cylinder about its axis and about one of its generators

125

(1)

The compound pendulum

125

(2)

Example: Circular cylinder oscillating about one of its generators

127

(1)

Motion about a fixed point

128

(2)

Inertia tensor

130

(3)

Example: Inertia tensor for four point masses in a plane

130

(1)

Example: Verification of the perpendicular axes theorem for a lamina using the inertia tensor

131

(1)

Example: Inertia tensor for a thin disc

132

(1)

Exercises

133

(2)

Non-inertial frames of reference

135

(12)

Coordinate system moving with constant linear acceleration

135

(2)

Example: A bucket of water undergoing constant acceleration

136

(1)

Coordinate system rotating with constant angular velocity

137

(8)

Rate of change of a vector relative to a rotating coordinate system

137

(1)

Interpretation of acceleration in a rotating frame

138

(1)

Newton's second law in a rotating frame: the centrifugal and Coriolis forces

139

(1)

Example: A bucket of water in a rotating frame

140

(1)

Example: A particle falling freely to earth

141

(2)

Example: Foucault's pendulum

143

(2)

Inertial and fictitious forces

145

(1)

Exercises

145

(2)

Variable-mass problems

147

(12)

Derivation of the general equation of motion from dp/dt = F

147

(1)

Accretion-of-mass problems

148

(4)

A raindrop falling through a cloud

149

(1)

Example: Falling raindrop with given rate of mass increase

150

(1)

Example: An open-topped railway wagon in a shower of rain

151

(1)

The rocket-propulsion equation

152

(6)

Example: A rocket with no external force acting

154

(1)

Example: Rocket taking off in a gravitational field

154

(1)

Multi-stage rockets

155

(2)

Example: Final speed of a two-stage rocket

157

(1)

Exercises

158

(1)

Collisions

159

(12)

Elastic collisions

159

(3)

Example: Elastic collision of identical particles: particles move off at right angles to each other

159

(2)

Example: Relationship between initial and final speeds of an elastically scattered particle

161

(1)

Conservation laws for a general two-body collision

162

(1)

LAB and centre-of-mass (CM) frames

163

(3)

Conservation of energy and momentum in the LAB frame

164

(1)

Conservation of energy and momentum in the CM frame

164

(2)

Galilean transformation between the Lab and CM frames

166

(4)

Relationship between &thetas;2 and &thetas;'

167

(1)

Relationship between &thetas;1 and &thetas;'

168

(1)

Relation between scattering angles in elastic collision of equal-mass particles as a special case

169

(1)

Exercises

170

(1)

Scattering of particles

171

(15)

Scattering from a fixed centre of force

172

(1)

Impact parameter and scattering angle

172

(2)

Differential and total cross-sections for scattering

174

(3)

Calculation of the differential cross-section in the CM frame

174

(2)

Transformation to the LAB frame

176

(1)

Hard-sphere scattering as a special case

177

(5)

Example: Scattering of a small sphere by a large, fixed sphere

177

(2)

Example: Scattering by identical spherical particles which are fixed in space

179

(2)

Example: Scattering by identical spherical particles which are free to recoil

181

(1)

Scattering in a central-force field

182

(1)

Repulsive inverse-square law of force

182

(1)

Rutherford's model of the atom

183

(1)

Attractive scattering forces and capture processes

183

(2)

Example: A planet passing through a cloud of meteors

184

(1)

Exercises

185

(1)

Special relativity

186

(18)

The need for a new principle of relativity

186

(1)

Einstein's axioms

187

(1)

Derivation of the Lorentz transformations

188

(4)

Implications of replacing Galilean with Lorentz invariance

192

(4)

Relativity of simultaneity

192

(1)

Lorentz--Fitzgerald contraction

193

(1)

Example: Rod at an angle to the x-axis moving in the x-direction

194

(1)

Example: Moving particle track at an angle to the x-axis in S relative to S'

195

(1)

Time dilation

196

(2)

Experimental evidence

197

(1)

Example: Decay of high-energy pions

197

(1)

The twins paradox

198

(2)

The experiment

199

(1)

The paradox or apparent contradiction

199

(1)

The resolution of the paradox

200

(1)

Example: Analogy of Lorentz transformation with rotations

200

(1)

General Lorentz transformation

201

(1)

Exercises

202

(2)

Relativistic kinematics

204

(18)

Lorentz transformation of intervals between events

204

(1)

Velocity transformations

205

(9)

Composition of velocities and verification of the Lorentz invariance of the speed of light

207

(1)

Example: A problem involving two spaceships and the Earth

207

(1)

Example: Another problem involving two spaceships and the Earth

208

(1)

Fresnel drag coefficient

209

(1)

Stellar aberration

210

(2)

Doppler effect

212

(1)

Thermal Doppler effect

213

(1)

Rapidity

214

(4)

Example: Lorentz transformations as hyperbolic functions

214

(1)

Example: Transitive property of the Lorentz transformations

215

(1)

Velocity composition in terms of rapidities

216

(1)

Example: Speed of multi-stage rocket relative to Earth

216

(1)

Example: Relationship between gamma factors for a rocket moving relative to S', where S' is moving relative to S

217

(1)

Acceleration transformation and proper acceleration

218

(1)

Apparent rotations and changes in shape

219

(1)

Exercises

220

(2)

Space--time geometry

222

(23)

Vector spaces: background and motivation

222

(3)

Four-space: the old-fashioned version

223

(1)

Four-space: the modern version

224

(1)

Lorentz-invariant interval between events in 4-space

225

(1)

Causality

225

(2)

The light cone

226

(1)

Four-vectors: spacelike, timelike and null

227

(2)

Scalar product of 4-vectors

227

(1)

Classification of 4-vectors

228

(1)

Proper time element dτ

229

(1)

Four-velocity of a body

230

(1)

Invariant scalar product

230

(1)

Four-acceleration of a body

231

(2)

Orthogonality of 4-velocity and 4-acceleration

231

(1)

Example: Express 4-acceleration in terms of 3-acceleration and 3-velocity

232

(1)

Transformation rules for 4-vectors

233

(4)

Example: Obtain 3-velocity transformation laws from the rules for 4-vectors

234

(1)

Example: Obtain linear acceleration in the comoving frame

235

(1)

Example: Central acceleration in the comoving frame

236

(1)

Usefulness of 4-vectors

237

(1)

Reconsideration of special relativity

237

(6)

The metric tensor

237

(1)

Example: Metric tensors for Euclidean 3-space and Minkowski 4-space

238

(2)

Geodesics (timelike and null)

240

(1)

Restatement of special relativity

241

(2)

Exercises

243

(2)

Relativistic dynamics

245

(25)

Particle collisions

245

(1)

Derivation of Lorentz-invariant 4-momentum conservation law

246

(1)

Relativistic momentum

246

(1)

Kinetic energy and the mass-energy relation

247

(1)

Momentum--energy relationship

248

(2)

Derivation from definitions of energy and momentum

248

(1)

Verification of Lorentz invariance

249

(1)

Example: Coalescence of two moving particles

249

(1)

Lorentz invariants

250

(1)

Massless particles

251

(4)

Example: Compton scattering using conservation of 4-momentum

252

(2)

Compton scattering: alternative calculation using Lorentz invariants

254

(1)

Examples of high-energy collisions

255

(5)

Example: Photon absorbed by stationary proton

256

(1)

Example: Two electrons scattering at right angles in the CM frame

257

(2)

Example: Proton makes head-on collision with photon

259

(1)

Example: Hard-sphere collisions revisited

260

(2)

Rockets revisited

262

(4)

The photon rocket

262

(1)

Example: Conventional rockets

263

(2)

Example: Comparison of conventional and photon rockets

265

(1)

Relativistic treatment of forces

266

(3)

The 4-vector form of N2

268

(1)

Exercises

269

(1)

Towards general relativity

270

(25)

Principle of equivalence

270

(4)

Einstein's lift experiments

271

(1)

Gravitation as an inertial force

272

(1)

The inertial frame revisited

273

(1)

Curved space

274

(4)

The necessity for a curved metric

274

(2)

Equivalence of a non-inertial frame and curved space

276

(1)

Signature of a vector space

277

(1)

Curved vector spaces

278

(4)

Geometry of space curves in 3D

278

(2)

Curvature in (1 + 3)D

280

(2)

The Einstein tensor

282

(1)

The field equations of general relativity

282

(4)

Einstein field equations for gravity

283

(2)

The full field equations of general relativity

285

(1)

Solving the field equations

286

(2)

The Schwarzschild metric

286

(1)

The matter tensor

287

(1)

The predictions of general relativity

288

(5)

Precession of the planetary orbits

288

(1)

The gravitational redshift

289

(1)

Black holes

290

(3)

Exercises

293

(2)

Appendices

295

(74)

A Integration using a dummy variable

297

(2)

B Solid angles

299

(2)

C Relativistic inertial mass

301

(1)

D The Penrose--Terrell rotation

302

(2)

E Solutions to exercises

304

(65)

Index

369

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Dynamics and Relativity

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Table of Contents

Supplemental Materials

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