Table of Contents
Preface v
1 Introduction 1
1.1 Calculus of Variations 1
1.2 Geodesics on a Sphere 2
1.3 Lagrangian Mechanics 4
1.4 Equivalence of Euler-Lagrange Equations to Newton's Second Law 6
1.5 Catenary 8
1.6 Brachistochrone 11
1.7 Exercises 13
2 One Degree of Freedom 23
2.1 Harmonic Oscillator; Damping 23
2.2 Another Example of One Dimensional Motion 25
2.3 Falling Object; Terminal Speed 27
2.4 Huygens' Pendulum 28
2.5 Plane Pendulum 30
2.6 Quartic (Duffing) Oscillator 33
2.7 van der Pol Equation 35
2.8 Inverse Problem 37
2.9 Linear Rocket Motion 39
2.10 Exercises 40
3 Systems with a Few Degrees of Freedom 53
3.1 Projectiles; Air Resistance 53
3.2 Spherical Pendulum 54
3.3 Two-dimensional Harmonic Motion 58
3.4 Planar Double Pendulum 61
3.5 Triple Pendulum 63
3.6 Central Forces 64
3.7 Holonomic and Non-holonomic Constraints 66
3.8 Exercises 68
4 Systems of Particles 79
4.1 Lagrangian for Several Particles; Noether's Theorem 79
4.2 Center of Mass 81
4.3 Rigid Bodies: Moment of Inertia 84
4.4 Euler Angles 88
4.5 Axially Symmetric Top 90
4.6 Euler Equations 91
4.7 Rotation about an Axis 93
4.8 Exercises 96
5 Noninertial Coordinate Systems 107
5.1 Accelerated Reference Frame 107
5.2 Rotating Frame 108
5.3 Bodies Falling to the Ground 110
5.4 Projectile Motion 111
5.5 Foucault Pendulum 113
5.6 Exercises 114
6 Gravitation 127
6.1 Kepler's Laws 127
6.2 Inverse Square Force 128
6.3 Two Body Problem 132
6.4 Restricted Three Body Problem 133
6.5 Stability of L4 and L5 136
6.6 Gravity from a Mass Distribution 137
6.7 Deviations from Newtonian Gravitation: Precession 138
6.8 The Tides 140
6.9 Exercises 141
7 Collisions and Scattering 155
7.1 Collinear, Elastic Collisions 155
7.2 Classical Zeno Process 157
7.3 Non-collinear Elastic Collisions 159
7.4 Scattering from Hard Sphere 161
7.5 Scattering by Central Potential 162
7.6 Rutherford Scattering 164
7.7 Cross Section for Repulsive 1/r4 Potential 165
7.8 Inverse Scattering Problem 169
7.9 Exercises 173
8 Hamiltonian Mechanics 183
8.1 Hamilton's Equations 183
8.2 Poisson Brackets 185
8.3 Canonical Transformations 187
8.4 Hamilton Jacobi Equation 190
8.5 Maupertuis Principle 196
8.6 Adiabatic Invariance of Action 197
8.7 Exercises 198
9 Stability and Instability 213
9.1 Driven Inverted Pendulum 213
9.2 Harmonically Driven Inverted Pendulum 219
9.3 Driven Pendulum 222
9.4 Parametric Resonance 225
9.5 Driven Harmonic Oscillator 226
9.6 Driven Duffing Oscillator 230
9.7 Hénon-Heiles Model; KAM Theorem 232
9.8 Exercises 234
10 Continuous Systems 253
10.1 Symmetric Ring 253
10.2 Masses on String: Continuous Limit 255
10.3 Finite String 258
10.4 Non-uniform String 260
10.5 Exercises 261
Appendix Supplementary Topics 271
A.1 Special Functions 271
A.2 Some Principal Figures in Mechanics 274
Bibliography 275
Index 277
