Analytical mechanics

590 Pages·1998·38.25 MB·English

by  Louis N. Hand

Analytical
, Mechanics

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LOUIS N. HAND
JANET D. FINCH
ANALYTICAL MECHANICS

Analytical Mechanics provides a detailed introduction to the key analytical techniques of
classical mechanics, one of the cornerstones of physics. It deals with all the important
subjects encountered in an undergraduate course and prepares the reader thoroughly for
further study at the graduate level.

The authors set out the fundamentals of Lagrangian and Hamiltonian mechanics early
on in the book and go on to cover such topics as linear oscillators, planetary orbits, rigid~
body motion, small vibrations, nonlinear dynamics, chaos, and special relativity. A special
feature is the inclusion of many “e-mail questions,” which are intended to facilitate dialogue
between the student and instructor.

Many worked examples are given, and there are 250 homework exercises to help
students gain confidence and proficiency in problem solving. It is an idea) textbook for
undergraduate courses in classical mechanics and provides a sound foundation for graduate
study.

Louis N. Hand was educated at Swarthmore College and Stanford University. After
serving as an assistant professor at Harvard University during the 1964 academic year, he
came to the Physics Department of Cornell University where he has remained ever since.
He is presently researching in the field of accelerator physics.

Janet D, Finch, teaching associate in the Physics Department of Cornell University,
earned her BS in engineering physics from the University of Illinois, and her MS in
theoretical physics and her MA in teaching from Cornell. In 1994 she began working with
Professor Hand on the Classical Mechanics course from which this book developed. She
was the e-mail tutor for the course during the first-time implementation of this innovation.
ANALYTICAL MECHANICS

LOUIS N. HAND
and

JANET D. FINCH

be a 9 CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
The Pitt Building, Trumpington Street, Cambridge CB2 IRP, United Kingdom

CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building, Cambridge CB2 2RU, UK http: //www.cup.cam.ac.uk
40 West 20th Street, New York, NY 10011-4211, USA http: //www.cup.org

10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1998
This book is in copyright. Subject to statutory exception

and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without

the written permission of Cambridge University Press.

First published 1998

Printed in the United States of America

Typeset in Times Roman 10.75/14 in TEX 2e(TB}

A catalog record for this book is available from
the British Library
Library of Congress Cataloging in Publication Data

Hand, Louis N., 1933—
Analytical mechanics/Louis N. Hand, Janet D. Finch.

p. cm.
Includes bibliographical references and index.
ISBN 0 521 57327 O hardback
ISBN 0 521 57572 9 paperback
1. Mechanics, Analytic. 1. Finch, Janet D., 1969-

Il. Title.

QA805.H26 1998

531’,01'515352-de21 97-43334

crip

ISBN 0 521 57327 Ohardback
ISBN 0 521 57572 9 paperback
CONTENTS

Preface

1 LAGRANGIAN MECHANICS

Ll

1.2
13
14

15
1.6
1.7

18
19
1.10
ll
1.12
1.13

Example and Review of Newton's Mechanics: A Block Sliding on
an Inclined Plane

Using Virtual Work to Solve the Same Problem

Solving for the Motion of a Heavy Bead Sliding on a Rotating Wire
Toward a General Formula: Degrees of Freedom and Types

of Constraints

Generalized Velocities: How to “Cancel the Dots”

Virtual Displacements and Virtual Work — Generalized Forces
Kinetic Energy as a Function of the Generalized Coordinates

and Velocities

Conservative Forces: Definition of the Lagrangian L

Reference Frames

Definition of the Hamiltonian

How to Get Rid of Ignorable Coordinates

Discussion and Conclusions — What's Next after You Get the EOM?
An Example of a Solved Problem

Summary of Chapter I

Problems

Appendix A. About Nonholonomic Constraints

Appendix B. More about Conservative Forces

2 VARIATIONAL CALCULUS AND ITS APPLICATION TO MECHANICS

21
2.2
2.3
24
2.5
2.6

History

The Euler Equation

Relevance to Mechanics

Systems with Several Degrees of Freedom

Why Use the Variational Approach in Mechanics?
Lagrange Multipliers

xi
vi

Solving Problems with Explicit Holonomic Constraints

Nonintegrable Nonholonomic Constraints —- A Method that Works

Postscript on the Euler Equation with More Than

One Independent Variable

Summary of Chapter 2

Problems

Appendix. About Maupertuis and What Came to Be Called
“Maupertuis’ Principle”

3 LINEAR OSCILLATORS

3.1
3.2
3.3
3.4
3.5
3.6
37
3.8
3.9

Stable or Unstable Equilibrium?

Simple Harmonic Oscillator

Damped Simple Harmonic Oscillator (DSHO)

An Oscillator Driven by an External Force

Driving Force Is a Step Function

Finding the Green's Function for the SHO

Adding up the Delta Functions — Solving the Arbitary Force
Driving an Oscillator in Resonance

Relative Phase of the DSHO Oscillator with Sinusoidal Drive
Summary of Chapter 3

Problems

4 ONE-DIMENSIONAL SYSTEMS: CENTRAL FORCES AND
THE KEPLER PROBLEM

41
4.2
43
44
45
46
47

The Motion of a “Generic” One-Dimensional System
The Grandfather’s Clock

The History of the Kepler Problem

Solving the Central Force Problem

The Special Case of Gravitational Attraction
Interpretation of Orbits

Repulsive 4 Forces

Summary of Chapter 4

Problems

Appendix. Tables of Astrophysical Data

5 NOETHER’S THEOREM AND HAMILTONIAN DYNAMICS

5.1

5.2
5.3
5.4
55
5.6
57

Discovering Angular Momentum Conservation from
Rotational Invariance

Noether’s Theorem

Hamiltonian Dynamics

The Legendre Transformation

Hamilton's Equations of Motion

Liouville’s Theorem

Momentum Space
58

Hamiltonian Dynamics in Accelerated Systems
Summary of Chapter 5

Problems

Appendix A. A General Proof of Liouville’s Theorem
Using the Jacobian

Appendix B. Poincaré Recurrence Theorem

6 THEORETICAL MECHANICS: FROM CANONICAL
TRANSFORMATIONS TO ACTION-ANGLE VARIABLES

6.1
62
6.3
64
6.5
6.6
67

Canonical Transformations

Discovering Three New Forms of the Generating Function
Poisson Brackets

Hamilton-Jacobi Equation

Action-Angle Variables for 1-D Systems

Integrable Systems

Invariant Tori and Winding Numbers

Summary of Chapter 6

Problems

Appendix. What Does “Symplectic” Mean?

7 ROTATING COORDINATE SYSTEMS

rl
72
73
74
1S
76
V7
78
19
7.10

What Is a Vector?

Review: Infinitesimal Rotations and Angular Velocity
Finite Three-Dimensional Rotations
Rotated Reference Frames

Rotating Reference Frames

‘The Instantaneous Angular Velocity @
Fictitious Forces

The Tower of Pisa Problem

Why Do Hurricane Winds Rotate?
Foucault Pendulum

Summary of Chapter 7

Problems

8 THE DYNAMICS OF RIGID BODIES

8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9

Kinetic Energy of a Rigid Body

The Moment of Inertia Tensor

Angular Momentum of a Rigid Body

The Euler Equations for Force-Free Rigid Body Motion

Motion of a Torque-Free Symmetric Top

Force-Free Precession of the Earth: The “Chandler Wobble”
Definition of Euler Angles

Finding the Angular Velocity

Motion of Torque-Free Asymmetric Tops: Poinsot Construction

vii

207

213
217
218

254

259

267
267
271
272
275
276

283
284
286
291
292
293
299
300
304
305
viii CONTENTS

8.10 The Heavy Symmetric Top 313
8.11 Precession of the Equinoxes 317
8.12 Mach’s Principle 323
Summary of Chapter 8 325
Problems 326
Appendix A. What Is a Tensor? ' 333
Appendix B, Symmetric Matrices Can Always Be Diagonalized
by “Rotating the Coordinates” 336
Appendix C. Understanding the Earth’s Equatorial Bulge 339
9 THE THEORY OF SMALL VIBRATIONS 343
9.1 Two Coupled Pendulums 344
9.2 Exact Lagrangian for the Double Pendulum 348
9.3 Single Frequency Solutions to Equations of Motion 352
9.4 Superimposing Different Modes; Complex Mode Amplitudes 355
9.5 Linear Triatomic Molecule 360
9.6 Why the Method Always Works 363
9.7 _N Point Masses Connected by a String 367
Summary of Chapter 9 371
Problems 373
Appendix. What Is a Cofactor? 380
10 APPROXIMATE SOLUTIONS TO NONANALYTIC PROBLEMS 383
10.1 Stability of Mechanical Systems 384
10.2 Parametric Resonance 388
10.3 Lindstedt—Poincaré Perturbation Theory 398
10.4 Driven Anharmonic Oscillator 401
Summary of Chapter 10 4
Problems 413
WW CHAOTIC DYNAMICS 423
11.1 Conservative Chaos - The Double Pendulum: A Hamiltonian
System with Two Degrees of Freedom 426
11.2 The Poincaré Section 428
11.3. KAM Tori: The Importance of Winding Number 433
11.4 Irrational Winding Numbers 436
11.5 Poincaré-Birkhoff Theorem 439
11.6 Linearizing Near a Fixed Point: The Tangent Map and
the Stability Matrix 442
11.7 Following Unstable Manifolds: Homoclinic Tangles 446
11.8 Lyapunov Exponents 449
11.9 Global Chaos for the Double Pendulum 451
11.10. Effect of Dissipation 452

11.11 Damped Driven Pendulum 453
CONTENTS:

11.12 Fractals

11.13 Chaos in the Solar System
Student Projects
Appendix. The Logistic Map: Period-Doubling Route
to Chaos; Renormalization

12 SPECIAL RELATIVITY
12.1 Space-Time Diagrams
12.2. The Lorentz Transformation
12.3 Simuttaneity Is Relative
12.4 What Happens to y and z if We Move Parallel to the X Axis?
12.5 Velocity Transformation Rules
12.6 Observing Light Waves
12.7 What Is Mass?
12.8 Rest Mass Is a Form of Energy
12.9 How Does Momentum Transform?
12,10 More Theoretical “Evidence” for the Equivalence of Mass
and Energy
12.11 Mathematics of Relativity: Invariants and Four-Vectors
12,12 A Second Look at the Energy~Momentum Four-Vector
12.13 Why Are There Both Upper and Lower Greek Indices?
12.14 Relativistic Lagrangian Mechanics
12.15 What Is the Lagrangian in an Electromagnetic Field?
12.16 Does a Constant Force Cause Constant Acceleration?
12.17 Derivation of the Lorentz Force from the Lagrangian
12.18 Relativistic Circular Motion
Summary of Chapter 12
Problems
Appendix. The Twin Paradox
Bibliography
References
Index
PREFACE

PREREQUISITES

The Physics Department at Cornell offers two intermediate-level undergraduate
mechanics courses. This book evolved from lecture notes used in the more advanced of
the two courses. Most of the students who took this course were considering postgraduate
study leading to future careers in physics or astronomy. With a few exceptions, they had
previously taken an introductory honors course in mechanics at the level of Kleppner and
Kolenkow.* Many students also had an Advanced Placement physics course in high school.
Since we can assume that a solid background in introductory college-level physics already
exists, we have not included a systematic review of elementary mechanics in the book,
other than the brief example at the beginning of Chapter 1.

Familiarity with a certain few basic mathematical concepts is essential. The student
should understand Taylor series in more than one variable, partial derivatives, the chain rule,
and elementary manipulations with complex variables.' Some elementary knowledge of
matrices and determinants is also needed.* Almost all of the students who took the honors
analytic mechanics course at Cornell have either completed, or were concurrently registered
in, a mathematical physics course involving vector analysis, complex variable theory, and
techniques for solving ordinary and partial differential equations. However, a thorough
grounding in these subjects is not essential — in fact some of this material can be learned
by taking a course based on this book.

INTRODUCTION

Our intention in writing this book is to reduce the gap between undergraduate
and graduate physics training. Graduate students often complain that their undergraduate
training did not prepare them for the rigors of graduate school. For that reason we have

* An Introduction to Mechanics, D. Kleppner and R. J. Kolenkow, McGraw-Hill, 1973.

1 At the level of Advanced Calculus, 2d ed., W. Kaplan, Addison-Wesley, 1984.

+ For linear algebra, we recommend a book on the level of Linear Algebra with Applications, 24 edition,
S.J. Leon, Macmillan, 1986, or one of the many other suitable texts at the intermediate level