# Classical Mechanics - Department of Physics and Astronomy

252 Pages · 2003 · 2.89 MB · English

• ## Classical Mechanics - Department of Physics and Astronomy

Classical Mechanics

Joel A. Shapiro

April 21, 2003 i

Copyright C 1994, 1997 by Joel A. Shapiro

stored in a retrieval system, or transmitted in any form or by any

means, electronic, mechanical, photocopying, or otherwise, without the

prior written permission of the author.

This is a preliminary version of the book, not to be considered a

fully published edition. While some of the material, particularly the

(cid:12)rst four chapters, is close to readiness for a (cid:12)rst edition, chapters 6

and 7 need more work, and chapter 8 is incomplete. The appendices

are random selections not yet reorganized. There are also as yet few

exercises for the later chapters. The (cid:12)rst edition will have an adequate

set of exercises for each chapter.

The author welcomes corrections, comments, and criticism. ii Contents

1 Particle Kinematics 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Single Particle Kinematics . . . . . . . . . . . . . . . . . 4

1.2.1 Motion in con(cid:12)guration space . . . . . . . . . . . 4

1.2.2 Conserved Quantities . . . . . . . . . . . . . . . . 6

1.3 Systems of Particles . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 External and internal forces . . . . . . . . . . . . 10

1.3.2 Constraints . . . . . . . . . . . . . . . . . . . . . 14

1.3.3 Generalized Coordinates for Unconstrained Sys-

tems . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.4 Kinetic energy in generalized coordinates . . . . . 19

1.4 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.1 Dynamical Systems . . . . . . . . . . . . . . . . . 22

1.4.2 Phase Space Flows . . . . . . . . . . . . . . . . . 27

2 Lagrange’s and Hamilton’s Equations 37

2.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . 37

2.1.1 Derivation for unconstrained systems . . . . . . . 38

2.1.2 Lagrangian for Constrained Systems . . . . . . . 41

2.1.3 Hamilton’s Principle . . . . . . . . . . . . . . . . 46

2.1.4 Examples of functional variation . . . . . . . . . . 48

2.1.5 Conserved Quantities . . . . . . . . . . . . . . . . 50

2.1.6 Hamilton’s Equations . . . . . . . . . . . . . . . . 53

2.1.7 Velocity-dependent forces . . . . . . . . . . . . . 55

3 Two Body Central Forces 65

3.1 Reduction to a one dimensional problem . . . . . . . . . 65

iii iv CONTENTS

3.1.1 Reduction to a one-body problem . . . . . . . . . 66

3.1.2 Reduction to one dimension . . . . . . . . . . . . 67

3.2 Integrating the motion . . . . . . . . . . . . . . . . . . . 69

3.2.1 The Kepler problem . . . . . . . . . . . . . . . . 70

3.2.2 Nearly Circular Orbits . . . . . . . . . . . . . . . 74

3.3 The Laplace-Runge-Lenz Vector . . . . . . . . . . . . . . 77

3.4 The virial theorem . . . . . . . . . . . . . . . . . . . . . 78

3.5 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . 79

4 Rigid Body Motion 85

4.1 Con(cid:12)guration space for a rigid body . . . . . . . . . . . . 85

4.1.1 Orthogonal Transformations . . . . . . . . . . . . 87

4.1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Kinematics in a rotating coordinate system . . . . . . . . 94

4.3 The moment of inertia tensor . . . . . . . . . . . . . . . 98

4.3.1 Motion about a (cid:12)xed point . . . . . . . . . . . . . 98

4.3.2 More General Motion . . . . . . . . . . . . . . . . 100

4.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.1 Euler’s Equations . . . . . . . . . . . . . . . . . . 107

4.4.2 Euler angles . . . . . . . . . . . . . . . . . . . . . 113

4.4.3 The symmetric top . . . . . . . . . . . . . . . . . 117

5 Small Oscillations 127

5.1 Small oscillations about stable equilibrium . . . . . . . . 127

5.1.1 Molecular Vibrations . . . . . . . . . . . . . . . . 130

5.1.2 An Alternative Approach . . . . . . . . . . . . . . 137

5.2 Other interactions . . . . . . . . . . . . . . . . . . . . . . 137

5.3 String dynamics . . . . . . . . . . . . . . . . . . . . . . . 138

5.4 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . 143

6 Hamilton’s Equations 147

6.1 Legendre transforms . . . . . . . . . . . . . . . . . . . . 147

6.2 Variations on phase curves . . . . . . . . . . . . . . . . . 152

6.3 Canonical transformations . . . . . . . . . . . . . . . . . 153

6.4 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . 155

6.5 Higher Di(cid:11)erential Forms . . . . . . . . . . . . . . . . . . 160

6.6 The natural symplectic 2-form . . . . . . . . . . . . . . . 169 CONTENTS v

6.6.1 Generating Functions . . . . . . . . . . . . . . . . 172

6.7 Hamilton{Jacobi Theory . . . . . . . . . . . . . . . . . . 181

6.8 Action-Angle Variables . . . . . . . . . . . . . . . . . . . 185

7 Perturbation Theory 189

7.1 Integrable systems . . . . . . . . . . . . . . . . . . . . . 189

7.2 Canonical Perturbation Theory . . . . . . . . . . . . . . 194

7.2.1 Time Dependent Perturbation Theory . . . . . . 196

7.3 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . 198

7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 198

7.3.2 For a time-independent Hamiltonian . . . . . . . 198

7.3.3 Slow time variation in H(q;p;t) . . . . . . . . . . 200

7.3.4 Systems with Many Degrees of Freedom . . . . . 206

7.3.5 Formal Perturbative Treatment . . . . . . . . . . 209

7.4 Rapidly Varying Perturbations . . . . . . . . . . . . . . . 211

7.5 New approach . . . . . . . . . . . . . . . . . . . . . . . . 216

8 Field Theory 219

8.1 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . 225

A (cid:15) and cross products 229

ijk

A.1 Vector Operations . . . . . . . . . . . . . . . . . . . . . . 229

A.1.1 (cid:14) and (cid:15) . . . . . . . . . . . . . . . . . . . . . . 229

ij ijk

C Gradient in Spherical Coordinates 237 vi CONTENTS Chapter 1

Particle Kinematics

1.1 Introduction

Classicalmechanics, narrowlyde(cid:12)ned, istheinvestigationofthemotion

of systems of particles in Euclidean three-dimensional space, under the

inﬂuence ofspeci(cid:12)ed forcelaws, withthemotion’sevolutiondetermined

by Newton’s second law, a second order di(cid:11)erential equation. That

is, given certain laws determining physical forces, and some boundary

conditions onthepositionsoftheparticles atsomeparticular times, the

problem is to determine the positions of all the particles at all times.

We will be discussing motions under speci(cid:12)c fundamental laws of great

physical importance, such as Coulomb’s law for the electrostatic force

between charged particles. We will also discuss laws which are less

fundamental, because the motion under them can be solved explicitly,

allowingthemtoserveasveryusefulmodelsforapproximationstomore

complicated physical situations, or as a testbed for examining concepts

in an explicitly evaluatable situation. Techniques suitable for broad

classes of force laws will also be developed.

The formalism of Newtonian classical mechanics, together with in-

vestigations into the appropriate force laws, provided the basic frame-

work for physics from the time of Newton until the beginning of this

century. The systems considered had a wide range of complexity. One

might consider a single particle on which the Earth’s gravity acts. But

one could also consider systems as the limit of an in(cid:12)nite number of

1 2 CHAPTER 1. PARTICLE KINEMATICS

very small particles, with displacements smoothly varying in space,

which gives rise to the continuum limit. One example of this is the

consideration of transverse waves on a stretched string, in which every

point on the string has an associated degree of freedom, its transverse

displacement.

The scope of classical mechanics was broadened in the 19th century,

in order to consider electromagnetism. Here the degrees of freedom

were not just the positions in space of charged particles, but also other

quantities, distributed throughout space, such as the the electric (cid:12)eld

at each point. This expansion in the type of degrees of freedom has

continued, and now in fundamental physics one considers many degrees

of freedom which correspond to no spatial motion, but one can still

discuss the classical mechanics of such systems.

As a fundamental framework for physics, classical mechanics gave

way onseveral frontstomoresophisticated concepts intheearly1900’s.

Mostdramatically,quantummechanicshaschangedourfocusfromspe-

ci(cid:12)csolutionsforthedynamicaldegrees offreedomasafunctionoftime

to the wave function, which determines the probabilities that a system

have particular values of these degrees of freedom. Special relativity

not only produced a variation of the Galilean invariance implicit in

Newton’s laws, but also is, at a fundamental level, at odds with the

basic ingredient of classical mechanics | that one particle can exert

a force on another, depending only on their simultaneous but di(cid:11)erent

positions. Finally general relativity brought out the narrowness of the

assumption that the coordinates of a particle are in a Euclidean space,

indicating instead not only that on the largest scales these coordinates

describe a curved manifold rather than a ﬂat space, but also that this

geometry is itself a dynamical (cid:12)eld.

Indeed, most of 20th century physics goes beyond classical Newto-

nian mechanics in one way or another. As many readers of this book

expect to become physicists working at the cutting edge of physics re-

search, and therefore will need to go beyond classical mechanics, we

begin with a few words of justi(cid:12)cation for investing e(cid:11)ort in under-

standing classical mechanics.

First of all, classical mechanics is still very useful in itself, and not

just for engineers. Consider the problems (scienti(cid:12)c | not political)

that NASA faces if it wants to land a rocket on a planet. This requires 1.1. INTRODUCTION 3

an accuracy of predicting the position of both planet and rocket far

beyond what one gets assuming Kepler’s laws, which is the motion one

predicts by treating the planet as a point particle inﬂuenced only by

the Newtonian gravitational (cid:12)eld of the Sun, also treated as a point

particle. NASA must consider other e(cid:11)ects, and either demonstrate

that they are ignorable or include them into the calculations. These

include

(cid:15) multipole moments of the sun

(cid:15) forces due to other planets

(cid:15) e(cid:11)ects of corrections to Newtonian gravity due to general relativ-

ity

(cid:15) friction due to the solar wind and gas in the solar system

Learning how to estimate or incorporate such e(cid:11)ects is not trivial.

Secondly, classical mechanics is not a dead (cid:12)eld of research | in

fact, in the last two decades there has been a great deal of interest in

\dynamical systems". Attention has shifted from calculation of the or-

bit over (cid:12)xed intervals of time to questions of the long-term stability of

the motion. New ways of looking at dynamical behavior have emerged,

such as chaos and fractal systems.

Thirdly, thefundamentalconceptsofclassicalmechanicsprovidethe

conceptual framework of quantum mechanics. For example, although

the Hamiltonian and Lagrangian were developed as sophisticated tech-

niques for performing classical mechanics calculations, they provide the

basic dynamical objects of quantum mechanics and quantum (cid:12)eld the-

ory respectively. One view of classical mechanics is as a steepest path

approximation to the path integral which describes quantum mechan-

ics. This integral over paths is of a classical quantity depending on the

\action" of the motion.

So classical mechanics is worth learning well, and we might as well

jump right in.

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