Table of Contents
Preface vii
Glossary xvii
1 Kinematics 1
1.1 Vectors and all that 1
1.1.1 Cartesian coordinates 1
1.1.2 Scalar product 3
1.1.3 Vector product 5
1.1.4 Dyadic product 8
1.1.5 Three-vector product 8
1.1.6 Double vector product 9
1.1.7 Infinitesimal rotations 10
1.1.8 Finite rotations 13
1.1.9 An application: Motion on a circle 18
1.1.10 Polar coordinates, cylindrical coordinates 20
1.1.11 Spherical coordinates 28
1.2 Fields and their gradients 33
1.3 Surface and volume elements 37
1.3.1 Surface elements 37
1.3.2 Volume elements 39
2 Dynamics 41
2.1 Newton's equation of motion 41
2.2 Elementary examples 42
2.2.1 Force-free motion 42
2.2.2 Constant force 43
2.2.3 Frictional forces 44
2.2.4 Linear restoring force: Harmonic oscillations 48
2.2.5 Damped harmonic oscillations 51
2.2.6 Damped and driven harmonic oscillations 56
a Periodic harmonic oscillations 56
b Impulsive force: Heaviside's step function and Dirac's delta function 60
c Arbitrary driving force: Green's function 66
d Periodic impulsive force: Cyclically steady state 69
3 Conservative Forces 73
3.1 One-dimensional motion 73
3.1.1 Kinetic and potential energy 73
3.1.2 Bounded motion between two turning points 75
a Small-amplitude oscillations 76
b Large-amplitude oscillations 77
c Potential energy inferred from the energy-dependent period 79
3.1.3 Unbounded motion with a single turning point 82
3.1.4 Unbounded motion without a turning point 85
3.2 Three-dimensional motion 87
3.2.1 Kinetic energy, potential energy 87
3.2.2 Conservative force fields 89
a Necessity of vanishing curl 89
b Sufficiency of vanishing curl?Stokes's theorem 90
3.2.3 Extremal points of the potential energy: Maxima, minima, saddle points 98
3.2.4 Potential energy in the vicinity of an extremal point 99
3.2.5 Example: Electrostatic potentials have no maxima or minima 104
4 Pair Forces 107
4.1 Reciprocal forces: Conservation of momentum 107
4.2 Conservative pair forces: Conservation of energy 109
4.3 Line-of-sight forces: Conservation of angular momentum 111
4.4 Conservative line-of-sight forces 112
4.5 Additional external forces 113
4.5.1 Transfer of momentum, energy, and angular momentum 113
4.5.2 Center-of-mass motion 114
4.5.3 Conservative external forces 117
5 Two-Body Systems 121
5.1 Center-of-mass motion and relative motion, reduced mass 121
5.2 Kepler's ellipses and Newton's force law 124
5.3 Motion in a central-force field 132
5.3.1 Bounded motion 132
5.3.2 Unbounded motion 137
5.3.3 Scattering 139
6 Gravitating Mass Distributions 145
6.1 Gravitational potential 145
6.2 Monopole moment and quadrupole moment dyadic 149
6.3 Newton's shell theorem 151
6.4 Green's function of the Laplacian differential operator 155
7 Variational Problems 159
7.1 Johann Bernoulli's challenge: The brachistochrone 159
7.2 Euler-Lagrange equations 161
7.3 Solution of the brachistochrone problem 164
7.4 Jakob Bernoulli's problem: The catenary 166
7.5 Handling constraints: Lagrange multipliers 169
8 Principle of Stationary Action 173
8.1 Lagrange function 173
8.1.1 One coordinate 173
8.1.2 More coordinates 175
8.1.3 Change of description, cyclic coordinates 177
8.2 Time and energy 178
8.3 Examples 180
8.3.1 Two masses strung up 180
8.3.2 Two coupled harmonic oscillators 184
9 Small-Amplitude Oscillations 189
9.1 Near an equilibrium: Lagrange function and equations of motion 189
9.2 Characteristic frequencies and normal modes 192
9.3 Examples 194
9.3.1 Pendulum 194
9.3.2 Double pendulum 197
9.3.3 Linear triatomic molecule 200
10 From Lagrange to Hamilton 207
10.1 Time as a coordinate 207
10.2 Endpoint variations: Momentum and Hamilton function 210
10.3 Five remarks 213
10.3.1 Natural variables 213
10.3.2 The minus sign 213
10.3.3 Legendre transformations 214
10.3.4 Many coordinates 215
10.3.5 Kinetic and canonical momentum 215
10.4 Cyclic coordinates and constants of motion 217
10.4.1 Energy 218
10.4.2 Total momentum 219
10.4.3 Total angular momentum 221
10.5 Hamilton's equations of motion 222
10.6 Poisson bracket 226
10.7 Conservation laws and symmetries. Noether's theorem 228
10.8 One-dimensional motion; two-dimensional phase space 229
10.9 Phase-space density. Liouville's theorem 232
10.10 Velocity-dependent forces and Schwinger's action 236
10.11 An excursion into the quantum realm 240
11 Rigid Bodies 243
11.1 Inertia dyadic. Steiner's theorem. Principal axes 243
11.2 Euler's equation of motion 248
11.2.1 The general case 248
11.2.2 No torque acting 250
a Two equal moments of inertia 250
b Rotation about a principal axis 251
11.3 Examples 253
11.3.1 Physical pendulum 253
11.3.2 Thin rod 254
11.3.3 Symmetric top 258
12 Earth-Bound Laboratories 265
12.1 Coriolis force, centrifugal force 265
12.2 Examples 270
12.2.1 Foucault's pendulum 270
12.2.2 Deflection of a falling mass 273
12.2.3 Gyrocompass 277
Exercises with Hints 281
Exercises for Chapters 1-12 281
Hints 312
Appendix 323
A On conic sections 323
A.1 Foci; vertices; cartesian coordinates: ray optics 323
A.2 Eccentricity; directrices; polar coordinates 325
A.3 Plane sections of a cone 328
B On the exercise for the reader in Section 5.2 332
C On the exercise for the reader in Section 11.2.2 334
Index 339
