Table of Contents
Preface vii
1 The Calculus of Variations 1
1.1 Foundations of the Calculus of Variations 1
1.1.1 A Simple Minimization Problem 1
1.1.2 Methods of the Calculus of Variations 3
1.1.3 Path of Shortest Distance and Geodesic Equation 7
1.2 Classical Variational Problems 10
1.2.1 Isoperimetric Problem 10
1.2.2 Brachistochrone Problem 13
1.3 Fermat's Principle of Least Time 14
1.3.1 Light Propagation in a Nonuniform Medium 16
1.3.2 Snell's Law 19
1.3.3 Application of Fermat's Principle 19
1.4 Geometric Formulation of Ray Optics* 20
1.4.1 Frenet-Serret Curvature of Light Path 20
1.4.2 Light Propagation in Spherical Geometry 23
1.4.3 Geodesic Representation of Light Propagation 25
1.4.4 Wavefront Representation 28
1.5 Problems 29
2 Lagrangian Mechanics 35
2.1 Maupertuis-Jacobi Principle of Least Action 35
2.1.1 Maupertuis' Principle 36
2.1.2 Jacobi's Principle 37
2.2 d' Alembert's Principle 38
2.2.1 Principle of Virtual Work 39
2.2.2 Lagrange's Equations from d' Alembert's Principle 40
2.3 Hamilton's Principle 43
2.3.1 Constraint Forces 43
2.3.2 Generalized Coordinates in Configuration Space 44
2.3.3 Constrained Motion on a Surface 46
2.3.4 Euler-Lagrange Equations 47
2.3.5 Four-step Lagrangian Method 48
2.3.6 Lagrangian Mechanics in Curvilinear Coordinates* 49
2.4 Lagrangian Mechanics in Configuration Space 50
2.4.1 Example I: Pendulum 50
2.4.2 Example II: Bead on a Rotating Hoop 51
2.4.3 Example III: Rotating Pendulum 53
2.4.4 Example IV: Compound Atwood Machine 55
2.4.5 Example V: Pendulum with Oscillating Fulcrum 57
2.5 Symmetries and Conservation Laws 58
2.5.1 Energy Conservation Law 60
2.5.2Momentum Conservation Laws 60
2.5.3 Invariance Properties of a Lagrangian 60
2.5.4 Lagrangian Mechanics with Symmetries 62
2.5.5 Routh's Procedure 62
2.6 Lagrangian Mechanics in the CM Frame 64
2.7 Problems 66
3 Hamiltonian Mechanics 73
3.1 Canonical Hamilton's Equations 73
3.2 Legendre Transformation* 75
3.3 Hamiltonian Optics and Wave-Particle Duality* 76
3.4 Motion in an Electromagnetic Field 78
3.4.1 Euler-Lagrange Equations 78
3.4.2 Energy Conservation Law 79
3.4.3 Gauge Invariance 80
3.4.4 Canonical Hamilton's Equations 81
3.4.5 Maupertuis' Principle of Least Action 81
3.5 One-degree-of-freedom Hamiltonian Dynamics 82
3.5.1 Enery Method 82
3.5.2 Simple Harmonic Oscillator 85
3.5.3 Pendulum 85
3.5.4 Constrained Motion on the Surface of a Cone 89
3.6 Problems 90
4 Motion in a Central-Force Field 95
4.1 Motion in a Central-Force Field 95
4.1.1 Lagrangian Formalism 95
4.1.2 Hamiltonian Formalism 95
4.1.3 Turning Points 99
4.2 Homogeneous Central Potentials* 99
4.2.1 The Virial Theorem 99
4.2.2 General Properties of Homogeneous Potentials 101
4.3 Kepler Problem 101
4.3.1 Bounded Keplerian Orbits 103
4.3.2 Unbounded Keplerian Orbits 105
4.3.3 Laplace-Runge-Lenz Vector* 106
4.4 Isotropic Simple Harmonic Oscillator 108
4.5 Internal Reflection inside a Well 110
4.6 Problems 112
5 Collisions and Scattering Theory 119
5.1 Two-Particle Collisions in the LAB Frame 120
5.2 Two-Particle Collisions in the CM Frame 122
5.3 Connection between the CM and LAB Frames 123
5.4 Scattering Cross Sections 124
5.4.1 Definitions 125
5.4.2 Cross Sections in CM and LAB Frames 126
5.5 Rutherford Scattering 128
5.6 Hard-Sphere and Soft-Sphere Scattering 130
5.6.1 Hard-Sphere Scattering 130
5.6.2 Soft-Sphere Scattering 132
5.7 Elastic Scattering by a Hard Surface 134
5.8 Problems 136
6 Motion in a Non-Inertial Frame 141
6.1 Time Derivatives in Rotating Frames 141
6.2 Accelerations in Rotating Frames 143
6.3 Lagrangian Formulation of Non-Inertial Motion 144
6.4 Motion Relative to Earth 146
6.4.1 Free-Fall Problem Revisited 150
6.4.2 Foucault Pendulum 151
6.5 Problems 154
7 Rigid Body Motion 159
7.1 Inertia Tensor of a Rigid Body 159
7.1.1 Discrete Particle Distribution 159
7.1.2 Parallel-Axes Theorem 161
7.1.3 Continuous Particle Distribution 161
7.1.4 Principal Axes of Inertia 163
7.2 Eulerian Rigid-Body Dynamics 166
7.2.1 Euler Equations 166
7.2.2 Euler Equations for a Force-Free Symmetric Top 167
7.2.3 Euler Equations for a Force-Free Asymmetric Top 170
7.3 Lagrangian Rigid-Body Dynamics 172
7.3.1 Eulerian Angles as Generalized Coordinates 172
7.3.2 Angular Velocity in Terms of Eulerian Angles 173
7.3.3 Rotational Kinetic Energy of a Symmetric Top 174
7.3.4 Symmetric Top with One Fixed Point 175
7.3.5 Stability of the Sleeping Top 181
7.4 Problems 182
8 Normal-Mode Analysis 187
8.1 Stability of Equilibrium Points 187
8.1.1 Sleeping Top 187
8.1.2 Bead on a Rotating Hoop 188
8.1.3 Circular Orbits in Central-Force Fields 189
8.2 Small Oscillations about Stable Equilibria 190
8.3 Normal-Mode Analysis of Coupled Oscillations 192
8.3.1 Coupled Simple Harmonic Oscillators 192
8.3.2 Coupled Nonlinear Oscillators 195
8.4 Problems 197
9 Continuous Lagrangian Systems 203
9.1 Waves on a Stretched String 203
9.1.1 Waves Equation 203
9.1.2 Lagrangian Formulation 204
9.2 Variational Principle for Field Theory* 205
9.2.1 Lagrangian Formulation 206
9.2.2 Noether Method and Conservation Laws 207
9.3 Schroedinger's Equation* 209
9.4 Euler Equations for a Perfect Fluid 211
9.4.1 Lagrangian Formulation 212
9.4.2 Energy-Momentum Conservation Laws 213
9.5 Problems 214
Appendix A Basic Mathematical Methods 217
A.1 Frenet-Serret Formulas 217
A.1.1 General Formulas 217
A.1.2 Frenet-Serret Formulas for Helical Path 219
A.2 Linear Algebra 220
A.2.1 Matrix Algebra 220
A.2.2 Eigenvalue Analysis of a 2 × 2 Matrix 222
A.3 Numerical Analysis 226
Appendix B Elliptic Functions and Integrals* 229
B.1 Jacobi Elliptic Functions 229
B.1.1 Definitions and Notation 229
B.1.2 Motion in a Quartic Potential 233
B.2 Weierstrass Elliptic Functions 234
B.2.1 Definitions and Notation 234
B.2.2 Motion in a Cubic Potential 239
B.3 Connection between Elliptic Functions 241
Appendix C Noncanonical Hamiltonian Mechanics* 245
C.1 Differential Geometry 245
C.2 Lagrange and Poisson Tensors 247
C.3 Hamiltonian Perturbation Theory 249
Bibliography 255
Index 257
