Classical Mechanics - Shapiro

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  • Classical Mechanics

    Joel A. Shapiro

    April 21, 2003

  • iCopyright C 1994, 1997 by Joel A. ShapiroAll rights reserved. No part of this publication may be reproduced,stored in a retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, or otherwise, without theprior written permission of the author.

    This is a preliminary version of the book, not to be considered afully published edition. While some of the material, particularly therst four chapters, is close to readiness for a rst edition, chapters 6and 7 need more work, and chapter 8 is incomplete. The appendicesare random selections not yet reorganized. There are also as yet fewexercises for the later chapters. The rst edition will have an adequateset of exercises for each chapter.

    The author welcomes corrections, comments, and criticism.

  • ii

  • Contents

    1 Particle Kinematics 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Single Particle Kinematics . . . . . . . . . . . . . . . . . 4

    1.2.1 Motion in conguration space . . . . . . . . . . . 41.2.2 Conserved Quantities . . . . . . . . . . . . . . . . 6

    1.3 Systems of Particles . . . . . . . . . . . . . . . . . . . . . 91.3.1 External and internal forces . . . . . . . . . . . . 101.3.2 Constraints . . . . . . . . . . . . . . . . . . . . . 141.3.3 Generalized Coordinates for Unconstrained Sys-

    tems . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.4 Kinetic energy in generalized coordinates . . . . . 19

    1.4 Phase Space . . . . . . . . . . . . . . . . . . . . . . . . . 211.4.1 Dynamical Systems . . . . . . . . . . . . . . . . . 221.4.2 Phase Space Flows . . . . . . . . . . . . . . . . . 27

    2 Lagranges and Hamiltons Equations 372.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . 37

    2.1.1 Derivation for unconstrained systems . . . . . . . 382.1.2 Lagrangian for Constrained Systems . . . . . . . 412.1.3 Hamiltons Principle . . . . . . . . . . . . . . . . 462.1.4 Examples of functional variation . . . . . . . . . . 482.1.5 Conserved Quantities . . . . . . . . . . . . . . . . 502.1.6 Hamiltons Equations . . . . . . . . . . . . . . . . 532.1.7 Velocity-dependent forces . . . . . . . . . . . . . 55

    3 Two Body Central Forces 653.1 Reduction to a one dimensional problem . . . . . . . . . 65

    iii

  • iv CONTENTS

    3.1.1 Reduction to a one-body problem . . . . . . . . . 663.1.2 Reduction to one dimension . . . . . . . . . . . . 67

    3.2 Integrating the motion . . . . . . . . . . . . . . . . . . . 693.2.1 The Kepler problem . . . . . . . . . . . . . . . . 703.2.2 Nearly Circular Orbits . . . . . . . . . . . . . . . 74

    3.3 The Laplace-Runge-Lenz Vector . . . . . . . . . . . . . . 773.4 The virial theorem . . . . . . . . . . . . . . . . . . . . . 783.5 Rutherford Scattering . . . . . . . . . . . . . . . . . . . . 79

    4 Rigid Body Motion 854.1 Conguration space for a rigid body . . . . . . . . . . . . 85

    4.1.1 Orthogonal Transformations . . . . . . . . . . . . 874.1.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . 91

    4.2 Kinematics in a rotating coordinate system . . . . . . . . 944.3 The moment of inertia tensor . . . . . . . . . . . . . . . 98

    4.3.1 Motion about a xed point . . . . . . . . . . . . . 984.3.2 More General Motion . . . . . . . . . . . . . . . . 100

    4.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4.1 Eulers Equations . . . . . . . . . . . . . . . . . . 1074.4.2 Euler angles . . . . . . . . . . . . . . . . . . . . . 1134.4.3 The symmetric top . . . . . . . . . . . . . . . . . 117

    5 Small Oscillations 1275.1 Small oscillations about stable equilibrium . . . . . . . . 127

    5.1.1 Molecular Vibrations . . . . . . . . . . . . . . . . 1305.1.2 An Alternative Approach . . . . . . . . . . . . . . 137

    5.2 Other interactions . . . . . . . . . . . . . . . . . . . . . . 1375.3 String dynamics . . . . . . . . . . . . . . . . . . . . . . . 1385.4 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . 143

    6 Hamiltons Equations 1476.1 Legendre transforms . . . . . . . . . . . . . . . . . . . . 1476.2 Variations on phase curves . . . . . . . . . . . . . . . . . 1526.3 Canonical transformations . . . . . . . . . . . . . . . . . 1536.4 Poisson Brackets . . . . . . . . . . . . . . . . . . . . . . 1556.5 Higher Dierential Forms . . . . . . . . . . . . . . . . . . 1606.6 The natural symplectic 2-form . . . . . . . . . . . . . . . 169

  • CONTENTS v

    6.6.1 Generating Functions . . . . . . . . . . . . . . . . 1726.7 Hamilton{Jacobi Theory . . . . . . . . . . . . . . . . . . 1816.8 Action-Angle Variables . . . . . . . . . . . . . . . . . . . 185

    7 Perturbation Theory 1897.1 Integrable systems . . . . . . . . . . . . . . . . . . . . . 1897.2 Canonical Perturbation Theory . . . . . . . . . . . . . . 194

    7.2.1 Time Dependent Perturbation Theory . . . . . . 1967.3 Adiabatic Invariants . . . . . . . . . . . . . . . . . . . . 198

    7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 1987.3.2 For a time-independent Hamiltonian . . . . . . . 1987.3.3 Slow time variation in H(q; p; t) . . . . . . . . . . 2007.3.4 Systems with Many Degrees of Freedom . . . . . 2067.3.5 Formal Perturbative Treatment . . . . . . . . . . 209

    7.4 Rapidly Varying Perturbations . . . . . . . . . . . . . . . 2117.5 New approach . . . . . . . . . . . . . . . . . . . . . . . . 216

    8 Field Theory 2198.1 Noethers Theorem . . . . . . . . . . . . . . . . . . . . . 225

    A ijk and cross products 229A.1 Vector Operations . . . . . . . . . . . . . . . . . . . . . . 229

    A.1.1 ij and ijk . . . . . . . . . . . . . . . . . . . . . . 229

    B The gradient operator 233

    C Gradient in Spherical Coordinates 237

  • vi CONTENTS

  • Chapter 1

    Particle Kinematics

    1.1 Introduction

    Classical mechanics, narrowly dened, is the investigation of the motionof systems of particles in Euclidean three-dimensional space, under theinfluence of specied force laws, with the motions evolution determinedby Newtons second law, a second order dierential equation. Thatis, given certain laws determining physical forces, and some boundaryconditions on the positions of the particles at some particular times, theproblem is to determine the positions of all the particles at all times.We will be discussing motions under specic fundamental laws of greatphysical importance, such as Coulombs law for the electrostatic forcebetween charged particles. We will also discuss laws which are lessfundamental, because the motion under them can be solved explicitly,allowing them to serve as very useful models for approximations to morecomplicated physical situations, or as a testbed for examining conceptsin an explicitly evaluatable situation. Techniques suitable for broadclasses 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 thiscentury. The systems considered had a wide range of complexity. Onemight consider a single particle on which the Earths gravity acts. Butone could also consider systems as the limit of an innite 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 theconsideration of transverse waves on a stretched string, in which everypoint on the string has an associated degree of freedom, its transversedisplacement.

    The scope of classical mechanics was broadened in the 19th century,in order to consider electromagnetism. Here the degrees of freedomwere not just the positions in space of charged particles, but also otherquantities, distributed throughout space, such as the the electric eldat each point. This expansion in the type of degrees of freedom hascontinued, and now in fundamental physics one considers many degreesof freedom which correspond to no spatial motion, but one can stilldiscuss the classical mechanics of such systems.

    As a fundamental framework for physics, classical mechanics gaveway on several fronts to more sophisticated concepts in the early 1900s.Most dramatically, quantum mechanics has changed our focus from spe-cic solutions for the dynamical degrees of freedom as a function of timeto the wave function, which determines the probabilities that a systemhave particular values of these degrees of freedom. Special relativitynot only produced a variation of the Galilean invariance implicit inNewtons laws, but also is, at a fundamental level, at odds with thebasic ingredient of classical mechanics | that one particle can exerta force on another, depending only on their simultaneous but dierentpositions. Finally general relativity brought out the narrowness of theassumption that the coordinates of a particle are in a Euclidean space,indicating instead not only that on the largest scales these coordinatesdescribe a curved manifold rather than a flat space, but also that thisgeometry is itself a dynamical eld.

    Indeed, most of 20th century physics goes beyond classical Newto-nian mechanics in one way or another. As many readers of this bookexpect to become physicists working at the cutting edge of physics re-search, and therefore will need to go beyond classical mechanics, webegin with a few words of justication for investing eort in under-standing classical mechanics.

    First of all, classical mechanics is still very useful in itself, and notjust for engineers. Consider the problems (scientic | not political)that NASA faces if it wants to land a rocket on a planet. This requires

  • 1.1. INTRODUCTION 3