INTRODUCTION TO LAGRANGIAN AND HAMILTONIAN MECHANICS Alain J. Brizard

220 Pages·2005·1.52 MB·English

INTRODUCTION TO
LAGRANGIAN AND HAMILTONIAN
MECHANICS
Alain J. Brizard
Department of Chemistry and Physics
Saint Michael’s College, Colchester, VT 05439
November 4, 2005
i
Preface
The original purpose of the present lecture notes on Classical Mechanics was to sup-
plement the standard undergraduate textbooks (such as Marion and Thorton’s Classical
Dynamics of Particles and Systems) normally used for an intermediate course in Classi-
cal Mechanics by inserting a more general and rigorous introduction to Lagrangian and
Hamiltonian methods suitable for undergraduate physics students at sophomore and ju-
nior levels. The outcome of this effort is that the lecture notes are now meant to provide
a self-consistent introduction to Classical Mechanics without the need of any additional
material.
It is expected that students taking this course will have had a one-year calculus-based
introductory physics course followed by a one-semester course in Modern Physics. Ideally,
students should have completed their three-semester calculus sequence by the time they
enroll in this course and, perhaps, have taken a course in ordinary differential equations.
On the other hand, this course should be taken before a rigorous course in Quantum
Mechanics in order to provide students with a sound historical perspective involving the
connection between Classical Physics and Modern Physics. Hence, the second semester of
the sophomore year or the fall semester of the junior year provide a perfect niche for this
course.
The structure of the lecture notes presented here is based on achieving several goals.
As a first goal, I originally wanted to model these notes after the wonderful monograph
of Landau and Lifschitz on Mechanics, which is often thought to be too concise for most
undergraduate students. Oneof themany positivecharacteristicsof Landau and Lifschitz’s
Mechanics is that Lagrangian mechanics is introduced in its first chapter and not in later
chapters as is usually done in more standard textbooks used at the sophomore/junior
undergraduate level. Consequently, Lagrangian mechanics becomes the centerpiece of the
course and it provides a continous thread throughout the text.
As a second goal, the lecturenotes introduce severalnumericalinvestigations of dynam-
ical equations appearing throughout the text. These numerical investigations present an
interactive pedagogical approach, which should enable students to begin their own numer-
ical investigations. As a third goal, an attempt was made to introduce relevant historical
facts (whenever appropriate) about the pioneers of Classical Mechanics. Much of the his-
torical information included in the Notes is taken from Ren´e Dugas (History of Mechanics,
1955), Wolfgang Yourgrau and Stanley Mandelstam (Variational Principles in Dynamics
and Quantum Theory, 1968), or Cornelius Lanczos (The Variational Principles of Me-
chanics, 1970). In fact, from a pedagogical point of view, this historical perspective helps
educating undergraduate students in establishing the deep connections between Classical
and Quantum Mechanics, which are often ignored or even inverted (as can be observed
when undergraduate students are surprised to hear that Hamiltonians have an indepen-
dent classical existence). As a fourth and final goal, I wanted to keep the scope of these
ii
notes limited to a one-semester course in contrast to standard textbooks, which often in-
clude an extensive review of Newtonian Mechanics as well as additional material such as
Hamiltonian chaos.
The standard topics covered in these notes are listed in order as follows: Introduction
to the Calculus of Variations (Chapter 1), Lagrangian Mechanics (Chapter 2), Hamiltonian
Mechanics (Chapter 3), Motion in a Central Field (Chapter 4), Collisions and Scattering
Theory (Chapter 5), Motion in a Non-Inertial Frame (Chapter 6), Rigid Body Motion
(Chapter 7), Normal-Mode Analysis (Chapter 8), and Continuous Lagrangian Systems
(Chapter 9). Each chapter contains a problem set with variable level of difficulty; sections
identified with an asterisk may be omitted. Lastly, in order to ensure a self-contained
presentation, a summary of mathematical methods associated with linear algebra, and
trigonometic and elliptic functions is presented in an Appendix.
Contents
1 Introduction to 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 . . . . . . . . . . . . . . . . . 2
1.1.3 Path of Shortest Distance and Geodesic Equation . . . . . . . . . . 7
1.2 Classical Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Brachistochrone Problem . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Fermat’s Principle of Least Time . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Light Propagation in a Nonuniform Medium . . . . . . . . . . . . . 14
1.3.2 Snell’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Application of Fermat’s Principle . . . . . . . . . . . . . . . . . . . 17
1.4 Geometric Formulation of Ray Optics∗ . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Frenet-Serret Curvature of Light Path . . . . . . . . . . . . . . . . 19
1.4.2 Light Propagation in Spherical Geometry . . . . . . . . . . . . . . . 21
1.4.3 Geodesic Representation of Light Propagation* . . . . . . . . . . . 23
1.4.4 Eikonal Representation . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Lagrangian Mechanics 29
2.1 Maupertuis-Jacobi’s Principle of Least Action . . . . . . . . . . . . . . . . 29
2.1.1 Maupertuis’ principle . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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iv CONTENTS
2.1.2 Jacobi’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 D’Alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Lagrange’s Equations from D’Alembert’s Principle . . . . . . . . . . 34
2.3 Hamilton’s Principle and Euler-Lagrange Equations . . . . . . . . . . . . . 36
2.3.1 Constraint Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Generalized Coordinates in Configuration Space . . . . . . . . . . . 37
2.3.3 Constrained Motion on a Surface . . . . . . . . . . . . . . . . . . . 39
2.3.4 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.5 Lagrangian Mechanics in Curvilinear Coordinates∗ . . . . . . . . . . 42
2.4 Lagrangian Mechanics in Configuration Space . . . . . . . . . . . . . . . . 42
2.4.1 Example I: Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 Example II: Bead on a Rotating Hoop . . . . . . . . . . . . . . . . 44
2.4.3 Example III: Rotating Pendulum . . . . . . . . . . . . . . . . . . . 47
2.4.4 Example IV: Compound Atwood Machine . . . . . . . . . . . . . . 48
2.4.5 Example V: Pendulum with Oscillating Fulcrum . . . . . . . . . . . 50
2.5 Symmetries and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 Energy Conservation Law . . . . . . . . . . . . . . . . . . . . . . . 53
2.5.2 Momentum Conservation Law . . . . . . . . . . . . . . . . . . . . . 54
2.5.3 Invariance Properties of a Lagrangian . . . . . . . . . . . . . . . . . 54
2.5.4 Lagrangian Mechanics with Symmetries . . . . . . . . . . . . . . . 55
2.5.5 Routh’s Procedure for Eliminating Ignorable Coordinates . . . . . . 58
2.6 Lagrangian Mechanics in the Center-of-Mass Frame . . . . . . . . . . . . . 58
2.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Hamiltonian Mechanics 65
3.1 Canonical Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Legendre Transformation* . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Hamiltonian Optics and Wave-Particle Duality* . . . . . . . . . . . . . . . 68
CONTENTS v
3.4 Particle Motion in an Electromagnetic Field*. . . . . . . . . . . . . . . . . 69
3.4.1 Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . 69
3.4.2 Energy Conservation Law . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.3 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.4 Canonical Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . 72
3.5 One-degree-of-freedom Hamiltonian Dynamics . . . . . . . . . . . . . . . . 72
3.5.1 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 75
3.5.2 Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.3 Constrained Motion on the Surface of a Cone . . . . . . . . . . . . 79
3.6 Charged Spherical Pendulum in a Magnetic Field∗ . . . . . . . . . . . . . . 80
3.6.1 Lagrangian and Routhian . . . . . . . . . . . . . . . . . . . . . . . 80
3.6.2 Routh-Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . 82
3.6.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Motion in a Central-Force Field 89
4.1 Motion in a Central-Force Field . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.1 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.2 Hamiltonian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.3 Turning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
∗
4.2 Homogeneous Central Potentials . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 The Virial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.2 General Properties of Homogeneous Potentials . . . . . . . . . . . . 94
4.3 Kepler Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.1 Bounded Keplerian Orbits . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.2 Unbounded Keplerian Orbits. . . . . . . . . . . . . . . . . . . . . . 99
4.3.3 Laplace-Runge-Lenz Vector∗ . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Isotropic Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 101
4.5 Internal Reflection inside a Well . . . . . . . . . . . . . . . . . . . . . . . . 103
vi CONTENTS
4.6 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5 Collisions and Scattering Theory 109
5.1 Two-Particle Collisions in the LAB Frame . . . . . . . . . . . . . . . . . . 109
5.2 Two-Particle Collisions in the CM Frame . . . . . . . . . . . . . . . . . . . 111
5.3 Connection between the CM and LAB Frames . . . . . . . . . . . . . . . . 112
5.4 Scattering Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.2 Scattering Cross Sections in CM and LAB Frames . . . . . . . . . . 115
5.5 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.6 Hard-Sphere and Soft-Sphere Scattering . . . . . . . . . . . . . . . . . . . 119
5.6.1 Hard-Sphere Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.6.2 Soft-Sphere Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.7 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6 Motion in a Non-Inertial Frame 127
6.1 Time Derivatives in Fixed and Rotating Frames . . . . . . . . . . . . . . . 127
6.2 Accelerations in Rotating Frames . . . . . . . . . . . . . . . . . . . . . . . 129
6.3 Lagrangian Formulation of Non-Inertial Motion . . . . . . . . . . . . . . . 131
6.4 Motion Relative to Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.4.1 Free-Fall Problem Revisited . . . . . . . . . . . . . . . . . . . . . . 135
6.4.2 Foucault Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 Rigid Body Motion 141
7.1 Inertia Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1.1 Discrete Particle Distribution . . . . . . . . . . . . . . . . . . . . . 141
7.1.2 Parallel-Axes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.1.3 Continuous Particle Distribution . . . . . . . . . . . . . . . . . . . 144
7.1.4 Principal Axes of Inertia . . . . . . . . . . . . . . . . . . . . . . . . 146
CONTENTS vii
7.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.1 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.2 Euler Equations for a Force-Free Symmetric Top . . . . . . . . . . . 149
7.2.3 Euler Equations for a Force-Free Asymmetric Top . . . . . . . . . . 151
7.2.4 Hamiltonian Formulation of Rigid Body Motion . . . . . . . . . . . 153
7.3 Symmetric Top with One Fixed Point . . . . . . . . . . . . . . . . . . . . . 154
7.3.1 Eulerian Angles as generalized Lagrangian Coordinates . . . . . . . 154
7.3.2 Angular Velocity in terms of Eulerian Angles . . . . . . . . . . . . . 155
7.3.3 Rotational Kinetic Energy of a Symmetric Top . . . . . . . . . . . . 156
7.3.4 Lagrangian Dynamics of a Symmetric Top with One Fixed Point . . 157
7.3.5 Stability of the Sleeping Top . . . . . . . . . . . . . . . . . . . . . . 163
7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8 Normal-Mode Analysis 167
8.1 Stability of Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.1 Bead on a Rotating Hoop . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.2 Circular Orbits in Central-Force Fields . . . . . . . . . . . . . . . . 168
8.2 Small Oscillations about Stable Equilibria . . . . . . . . . . . . . . . . . . 169
8.3 Coupled Oscillations and Normal-Mode Analysis . . . . . . . . . . . . . . . 171
8.3.1 Coupled Simple Harmonic Oscillators . . . . . . . . . . . . . . . . . 171
8.3.2 Nonlinear Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . 175
8.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9 Continuous Lagrangian Systems 181
9.1 Waves on a Stretched String . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.1.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.1.2 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.1.3 Lagrangian Description for Waves on a Stretched String . . . . . . 182
9.2 General Variational Principle for Field Theory* . . . . . . . . . . . . . . . 183
viii CONTENTS
9.2.1 Action Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2.2 Noether Method and Conservation Laws . . . . . . . . . . . . . . . 184
9.3 Variational Principle for Schroedinger Equation* . . . . . . . . . . . . . . . 185
A Basic Mathematical Methods 189
A.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
A.1.1 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
A.1.2 Eigenvalue analysis of a 2×2 matrix . . . . . . . . . . . . . . . . . 191
A.2 Important Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
A.2.1 Trigonometric Functions and Integrals . . . . . . . . . . . . . . . . 195
A.2.2 Elliptic Functions and Integrals* . . . . . . . . . . . . . . . . . . . 197
B Notes on Feynman’s Quantum Mechanics 209
B.1 Feynman postulates and quantum wave function . . . . . . . . . . . . . . . 209
B.2 Derivation of the Schroedinger equation . . . . . . . . . . . . . . . . . . . . 210
Chapter 1
Introduction to the Calculus of
Variations
A wide range of equations in physics, from quantum field and superstring theories to
general relativity,from fluid dynamics to plasma physics and condensed-matter theory, are
derivedfrom action (variational) principles. The purpose of this chapter is to introduce the
methods of the Calculus of Variations that figure prominently in the formulation of action
principles in physics.
1.1 Foundations of the Calculus of Variations
1.1.1 A Simple Minimization Problem
It is a well-known fact that the shortest distance between two points in Euclidean (”flat”)
space is calculated along a straight line joining the two points. Although this fact is
intuitivelyobvious, we begin our discussion of the problem of minimizing certain integrals
in mathematics and physics with a search for an explicit proof. In particular, we prove
that the straight line y (x) = mx yields a path of shortest distance between the two points
0
(0,0) and (1,m) on the (x,y)-plane as follows.
First, we consider the length integral
(cid:1) (cid:2)
1
L[y] = 1+(y(cid:2))2dx, (1.1)
0
where y(cid:2) = y(cid:2)(x) and the notation L[y] is used to denote the fact that the value of the
integral (1.1) depends on the choice we make for the function y(x); we insist, however,
that the function y(x) satisfy the boundary conditions y(0) = 0 and y(1) = m. Next, by
1