Introduction to Lagrangian and Hamiltonian Mechanics - BRIZARD, Angielskie techniczne
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July 14, 2004
INTRODUCTION TO
LAGRANGIAN AND HAMILTONIAN
MECHANICS
Alain J. Brizard
Department of Chemistry and Physics
Saint Michael’s College, Colchester, VT 05439
Contents
1 Introduction to the Calculus of Variations
1
1.1 Fermat’s Principle of Least Time . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Euler’s First Equation . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Euler’s Second Equation . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Snell’sLaw ............................... 6
1.1.4 Application of Fermat’s Principle . . . . . . . . . . . . . . . . . . . 7
1.2 Geometric Formulation of Ray Optics . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Frenet-Serret Curvature of Light Path . . . . . . . . . . . . . . . . 9
1.2.2 Light Propagation in Spherical Geometry . . . . . . . . . . . . . . . 11
1.2.3 Geodesic Representation of Light Propagation . . . . . . . . . . . . 13
1.2.4 Eikonal Representation . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Brachistochrone Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Problems..................................... 18
2 Lagrangian Mechanics
21
2.1 Maupertuis-Jacobi Principle of Least Action . . . . . . . . . . . . . . . . . 21
2.2 Principle of Least Action of Euler and Lagrange . . . . . . . . . . . . . . . 23
2.2.1 Generalized Coordinates in Configuration Space . . . . . . . . . . . 23
2.2.2 Constrained Motion on a Surface . . . . . . . . . . . . . . . . . . . 24
2.2.3 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Lagrangian Mechanics in Configuration Space . . . . . . . . . . . . . . . . 27
2.3.1 Example I: Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 27
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2.3.2 Example II: Bead on a Rotating Hoop . . . . . . . . . . . . . . . . 28
2.3.3 Example III: Rotating Pendulum . . . . . . . . . . . . . . . . . . . 30
2.3.4 Example IV: Compound Atwood Machine . . . . . . . . . . . . . . 31
2.3.5 Example V: Pendulum with Oscillating Fulcrum . . . . . . . . . . . 33
2.4 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Energy Conservation Law . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.2 Momentum Conservation Law . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Invariance Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.4 Lagrangian Mechanics with Symmetries . . . . . . . . . . . . . . . 38
2.4.5 Routh’s Procedure for Eliminating Ignorable Coordinates . . . . . . 39
2.5 Lagrangian Mechanics in the Center-of-Mass Frame . . . . . . . . . . . . . 40
2.6 Problems..................................... 43
3 Hamiltonian Mechanics
45
3.1 Canonical Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Legendre Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Hamiltonian Optics and Wave-Particle Duality* . . . . . . . . . . . . . . . 48
3.4 Particle Motion in an Electromagnetic Field* . . . . . . . . . . . . . . . . . 49
3.4.1 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 Energy Conservation Law . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.3 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.4 Canonical Hamilton’s Equationss . . . . . . . . . . . . . . . . . . . 51
3.5 One-degree-of-freedom Hamiltonian Dynamics . . . . . . . . . . . . . . . . 52
3.5.1 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 53
3.5.2 Pendulum................................ 54
3.5.3 Constrained Motion on the Surface of a Cone . . . . . . . . . . . . 56
3.6 Charged Spherical Pendulum in a Magnetic Field* . . . . . . . . . . . . . . 57
3.6.1 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6.2 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . 59
CONTENTS
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3.6.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.7 Problems..................................... 65
4 Motion in a Central-Force Field
67
4.1 Motion in a Central-Force Field . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 Homogeneous Central Potentials* . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.1 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 General Properties of Homogeneous Potentials . . . . . . . . . . . . 72
4.3 KeplerProblem................................. 72
4.3.1 Bounded Keplerian Orbits . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Unbounded Keplerian Orbits . . . . . . . . . . . . . . . . . . . . . . 76
4.3.3 Laplace-Runge-Lenz Vector* . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Isotropic Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 78
4.5 Internal Reflection inside a Well . . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Problems..................................... 83
5 Collisions and Scattering Theory
85
5.1 Two-Particle Collisions in the LAB Frame . . . . . . . . . . . . . . . . . . 85
5.2 Two-Particle Collisions in the CM Frame . . . . . . . . . . . . . . . . . . . 87
5.3 Connection between the CM and LAB Frames . . . . . . . . . . . . . . . . 88
5.4 Scattering Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.2 Scattering Cross Sections in CM and LAB Frames . . . . . . . . . . 91
5.5 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Hard-Sphere and Soft-Sphere Scattering . . . . . . . . . . . . . . . . . . . 94
5.6.1 Hard-Sphere Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 95
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5.6.2 Soft-Sphere Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.7 Problems..................................... 99
6 Motion in a Non-Inertial Frame
103
6.1 Time Derivatives in Fixed and Rotating Frames . . . . . . . . . . . . . . . 103
6.2 Accelerations in Rotating Frames . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Lagrangian Formulation of Non-Inertial Motion . . . . . . . . . . . . . . . 106
6.4 Motion Relative to Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.4.1 Free-Fall Problem Revisited . . . . . . . . . . . . . . . . . . . . . . 111
6.4.2 Foucault Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.5 Problems..................................... 116
7 Rigid Body Motion
117
7.1 InertiaTensor.................................. 117
7.1.1 Discrete Particle Distribution . . . . . . . . . . . . . . . . . . . . . 117
7.1.2 Parallel-Axes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1.3 Continuous Particle Distribution . . . . . . . . . . . . . . . . . . . 120
7.1.4 Principal Axes of Inertia . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.2.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.2.2 Euler Equations for a Force-Free Symmetric Top . . . . . . . . . . . 125
7.2.3 Euler Equations for a Force-Free Asymmetric Top . . . . . . . . . . 127
7.3 Symmetric Top with One Fixed Point . . . . . . . . . . . . . . . . . . . . . 130
7.3.1 Eulerian Angles as generalized Lagrangian Coordinates . . . . . . . 130
7.3.2 Angular Velocity in terms of Eulerian Angles . . . . . . . . . . . . . 131
7.3.3 Rotational Kinetic Energy of a Symmetric Top . . . . . . . . . . . . 132
7.3.4 Lagrangian Dynamics of a Symmetric Top with One Fixed Point . . 133
7.3.5 Stability of the Sleeping Top . . . . . . . . . . . . . . . . . . . . . . 139
7.4 Problems..................................... 140
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