Classical Mechanics: Point Particles and Relativity (Classical Theoretical Physics)

499 Pages·2003·2.921 MB·English

by  Walter Greiner

C M
LASSICAL ECHANICS
Point Particles and Relativity
Springer
NewYork
Berlin
Heidelberg
HongKong
London
Milan
Paris
Tokyo
Walter Greiner
C
LASSICAL
M
ECHANICS
Point Particles
and Relativity
Foreword by D. Allan Bromley
With 317 Figures
Springer
WalterGreiner
Institutfu¨rTheoretischePhysik
JohannWolfgangGoethe-Universita¨t
RobertMayerStrasse10
Postfach111932
D-60054FrankfurtamMain
Germany
[email protected]
LibraryofCongressCataloging-in-PublicationData
Greiner,Walter,1935-
Classicalmechanics:pointparticlesandrelativity/WalterGreiner.
p.cm.--(Classicaltheoreticalphysics)
Includesbibliographicalreferencesandindex.
ISBN0-387-95586-0(softcover:alk.paper)
1.Mechanics--Problems,exercises,etc. 2.Relativity(Physics)--Problems,exercises,
etc. I.Title II.Series.
QC125.2.G742003
531--dc21 2002030570
ISBN0-387-95586-0 Printedonacid-freepaper.
TranslatedfromtheGermanMechanik:Teil2,byWalterGreiner,publishedbyVerlagHarriDeutsch,Thun,
(cid:1)
FrankfurtamMain,Germany, c 1989.
(cid:1)
c 2004Springer-VerlagNewYork,Inc.
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermission
ofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010,USA),exceptforbrief
excerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinformationstorage
andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownor
hereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenot
identifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietary
rights.
PrintedintheUnitedStatesofAmerica.
9 8 7 6 5 4 3 2 1 SPIN10892857
www.springer-ny.com
Springer-Verlag NewYork Berlin Heidelberg
AmemberofBertelsmannSpringerScience+BusinessMediaGmbH
Foreword
MorethanagenerationofGerman-speakingstudentsaroundtheworldhaveworkedtheir
waytoanunderstandingandappreciationofthepowerandbeautyofmoderntheoretical
physics—with mathematics, the most fundamental of sciences—using Walter Greiner’s
textbooksastheirguide.
Theideaofdevelopingacoherent,completepresentationofanentirefieldofscienceina
seriesofcloselyrelatedtextbooksisnotanewone.Manyolderphysiciansrememberwith
realpleasuretheirsenseofadventureanddiscoveryastheyworkedtheirwaysthroughthe
classicseriesbySommerfeld,byPlanck,andbyLandauandLifshitz.Fromthestudents’
viewpoint, there are a great many obvious advantages to be gained through the use of
consistentnotation,logicalorderingoftopics,andcoherenceofpresentation;beyondthis,
thecompletecoverageofthescienceprovidesauniqueopportunityfortheauthortoconvey
hispersonalenthusiasmandloveforhissubject.
Thesevolumesonclassicalphysics,finallyavailableinEnglish,complementGreiner’s
textsonquantumphysics,mostofwhichhavebeenavailabletoEnglish-speakingaudiences
forsometime.Thecompletesetofbookswillthusprovideacoherentviewofphysicsthat
includes,inclassicalphysics,thermodynamicsandstatisticalmechanics,classicaldynam-
ics,electromagnetism,andgeneralrelativity;andinquantumphysics,quantummechanics,
symmetries,relativisticquantummechanics,quantumelectro-andchromodynamics,and
thegaugetheoryofweakinteractions.
WhatmakesGreiner’svolumesofparticularvaluetothestudentandprofessoralikeis
theircompleteness.Greineravoidsthealltoocommon“itfollowsthat...,”whichconceals
severalpagesofmathematicalmanipulationandconfoundsthestudent.Hedoesnothesitate
toincludeexperimentaldatatoilluminateorillustrateatheoreticalpoint,andthesedata,
likethetheoreticalcontent,havebeenkeptuptodateandtopicalthroughfrequentrevision
andexpansionofthelecturenotesuponwhichthesevolumesarebased.
Moreover,Greinergreatlyincreasesthevalueofhispresentationbyincludingsomething
like one hundred completely worked examples in each volume. Nothing is of greater
importance to the student than seeing, in detail, how the theoretical concepts and tools
v
vi FOREWORD
understudyareappliedtoactualproblemsofinteresttoworkingphysicists.And,finally,
Greiner adds brief biographical sketches to each chapter covering the people responsible
for the development of the theoretical ideas and/or the experimental data presented. It
wasAugusteComte(1789–1857)inhisPositivePhilosophywhonoted,“Tounderstanda
scienceitisnecessarytoknowitshistory.”Thisisalltoooftenforgotteninmodernphysics
teaching,andthebridgesthatGreinerbuildstothepioneeringfiguresofourscienceupon
whoseworkwebuildarewelcomeones.
Greiner’s lectures, which underlie these volumes, are internationally noted for their
clarity,fortheircompleteness,andfortheeffortthathehasdevotedtomakingphysicsan
integral whole. His enthusiasm for his sciences is contagious and shines through almost
everypage.
These volumes represent only a part of a unique and Herculean effort to make all of
theoreticalphysicsaccessibletotheinterestedstudent.Beyondthat,theyareofenormous
value to the professional physicist and to all others working with quantum phenomena.
Againandagain,thereaderwillfindthat,afterdippingintoaparticularvolumetoreviewa
specifictopic,heorshewillendupbrowsing,caughtupbyoftenfascinatingnewinsights
anddevelopmentswithwhichheorshehadnotpreviouslybeenfamiliar.
Having used a number of Greiner’s volumes in their original German in my teaching
and research at Yale, I welcome these new and revised English translations and would
recommendthementhusiasticallytoanyonesearchingforacoherentoverviewofphysics.
D.AllanBromley
HenryFordIIProfessorofPhysics
YaleUniversity
NewHaven,Connecticut,USA
Preface
Theoreticalphysicshasbecomeamanyfacetedscience.Fortheyoungstudent,itisdifficult
enough to cope with the overwhelming amount of new material that has to be learned,
let alone obtain an overview of the entire field, which ranges from mechanics through
electrodynamics,quantummechanics,fieldtheory,nuclearandheavy-ionscience,statistical
mechanics,thermodynamics,andsolid-statetheorytoelementary-particlephysics;andthis
knowledgeshouldbeacquiredinjusteighttotensemesters,duringwhich,inaddition,a
diplomaormaster’sthesishastobeworkedonorexaminationspreparedfor.Allthiscanbe
achievedonlyiftheuniversityteachershelptointroducethestudenttothenewdisciplines
asearlyaspossible,inordertocreateinterestandexcitementthatinturnsetfreeessential
newenergy.
At the Johann Wolfgang Goethe University in Frankfurt am Main, we therefore con-
front the student with theoretical physics immediately, in the first semester. Theoretical
MechanicsIandII,Electrodynamics,andQuantumMechanicsI—AnIntroductionarethe
coursesduringthefirsttwoyears.Theselecturesaresupplementedwithmanymathemati-
calexplanationsandmuchsupportmaterial.Afterthefourthsemesterofstudies,graduate
work begins, and Quantum Mechanics II—Symmetries, Statistical Mechanics and Ther-
modynamics,RelativisticQuantumMechanics,QuantumElectrodynamics,GaugeTheory
of Weak Interactions, and Quantum Chromodynamics are obligatory. Apart from these,
a number of supplementary courses on special topics are offered, such as Hydrodynam-
ics,ClassicalFieldTheory,SpecialandGeneralRelativity,Many-BodyTheories,Nuclear
Models,ModelsofElementaryParticles,andSolid-StateTheory.
Thisvolumeoflectures,ClassicalMechanics:PointParticlesandRelativity,dealswith
thefirstandmoreelementarypartoftheimportantfieldofclassicalmechanics.Wehave
tried to present the subject in a manner that is both interesting to the student and easily
accessible.Themaintextisthereforeaccompaniedbymanyexercisesandexamplesthat
havebeenworkedoutingreatdetail.Thisshouldmakethebookusefulalsoforstudents
wishingtostudythesubjectontheirown.
Beginningtheeducationintheoreticalphysicsatthefirstuniversitysemester,andnotas
dictatedbytraditionafterthefirstoneandahalfyearsinthethirdorfourthsemester,has
broughtalongquiteafewchangesascomparedtothetraditionalcoursesinthatdiscipline.
vii
viii PREFACE
Especiallynecessaryisagreateramalgamationbetweentheactualphysicalproblemsand
the necessary mathematics. Therefore, we treat in the first semester vector algebra and
analysis, the solution of ordinary, linear differential equations, Newton’s mechanics of a
mass point culminating in the discussion of Kepler’s laws (planetary motion), elements
of astronomy, addressing modern research issues like the dark matter problem, and the
mathematicallysimplemechanicsofspecialrelativity.
Many explicitly worked-out examples and exercises illustrate the new concepts and
methodsanddeepentheinterrelationshipbetweenphysicsandmathematics.Asamatterof
fact,thisfirst-semestercourseintheoreticalmechanicsisaprecursortotheoreticalphysics.
Thischangessignificantlythecontentofthelecturesofthesecondsemesteraddressedin
thevolumeClassicalMechanics:SystemofParticlesandHamiltonianDynamics.
Thenewmathematicaltoolsareexplainedandexercisedinmanyphysicalexamples.In
thelecturingpraxis,thedeepeningoftheexhibitedmaterialiscarriedoutinathree-hour-
per-week theoretica, that is, group exercises where eight or ten students solve the given
exercisesundertheguidanceofatutor.
Biographicalandhistoricalfootnotesanchorthescientificdevelopmentwithinthegeneral
context of scientific progress and evolution. In this context, I thank the publishers Harri
Deutsch and F. A. Brockhaus (Brockhaus Enzyklopa¨die, F.A. Brockhaus, Wiesbaden—
marked by [BR]) for giving permission to extract the biographical data of physicists and
mathematiciansfromtheirpublications.
We should also mention that in preparing some early sections and exercises of our
lectureswereliedonthebookTheoryandProblemsofTheoreticalMechanics,byMurray
R.Spiegel,McGraw-Hill,NewYork,1967.
Overtheyears,weenjoyedthehelpofseveralstudentsandcollaborators,inparticular,
H.Angermu¨ller,P.Bergmann,H.Betz,W.Betz,G.Binnig(Nobelprize1986),J.Briechle,
M.Bundschuh,W.Caspar,C.v.Charewski,J.v.Czarnecki,R.Fickler,R.Fiedler,B.Fricke
(nowprofessoratKasselUniversity),C.Greiner(nowprofessoratJWG-University,Frank-
furt am Main), M. Greiner, W. Grosch, R. Heuer, E. Hoffmann, L. Kohaupt, N. Krug,
P. Kurowski, H. Leber, H. J. Lustig, A. Mahn, B. Moreth, R. Mo¨rschel, B. Mu¨ller (now
professor at Duke University, Durham, N.C.), H. Mu¨ller, H. Peitz, J. Rafelski (now pro-
fessor at University of Arizona, Tuscon), G. Plunien, J. Reinhardt, M. Rufa, H. Schaller,
D.Schebesta,H.J.Scheefer,H.Schwerin,M.Seiwert,G.Soff(nowprofessoratTechnical
UniversityDresden),M.Soffel(nowprofessoratTechnicalUniversityDresden),E.Stein
(nowprofessoratMaharishiUniversity,Vlodrop,Netherlands),K.E.Stiebig,E.Sta¨mmler,
H.Stock,H.Sto¨rmer(Nobelprize1998),J.Wagner,andR.Zimmermann.Theyallmade
their way in science and society, and meanwhile work as professors at universities, as
leaders in industry, and in other places. We particularly acknowledge the recent help of
Dr.SvenSoffandDr.StefanSchererduringthepreparationoftheEnglishmanuscript.The
figuresweredrawnbyMrs.A.Steidl.
TheEnglishmanuscriptwascopy-editedbyKristenCassereauandtheproductionofthe
bookwassupervisedbyTimothyTaylorofSpringer-VerlagNewYork,Inc.
WalterGreiner
JohannWolfgangGoethe-Universita¨t
FrankfurtamMain,Germany
Contents
Foreword v
Preface vii
I VECTOR CALCULUS 1
1 IntroductionandBasicDefinitions 2
2 TheScalarProduct 5
3 ComponentRepresentationofaVector 9
4 TheVectorProduct(AxialVector) 13
5 TheTripleScalarProduct 25
6 ApplicationofVectorCalculus 27
Applicationinmathematics: 27
Applicationinphysics: 31
7 DifferentiationandIntegrationofVectors 39
8 TheMovingTrihedral(AccompanyingDreibein)—theFrenet
Formulas 49
ExamplesonFrenet’sformulas: 55
ix
x CONTENTS
9 SurfacesinSpace 64
10 CoordinateFrames 68
11 VectorDifferentialOperations 83
Theoperationsgradient,divergence,andcurl(rotation) 83
Differentialoperatorsinarbitrarygeneral(curvilinear)coordinates 96
12 DeterminationofLineIntegrals 109
13 TheIntegralLawsofGaussandStokes 112
GaussLaw: 112
TheGausstheorem: 114
GeometricinterpretationoftheGausstheorem: 115
Stokeslaw: 117
14 CalculationofSurfaceIntegrals 125
15 Volume(Space)Integrals 130
II NEWTONIAN MECHANICS 133
16 Newton’sAxioms 134
17 BasicConceptsofMechanics 140
Inertialsystems 140
Measurementofmasses 141
Work 141
Kineticenergy 142
Conservativeforces 142
Potential 143
Energylaw 144
Equivalenceofimpulseofforceandmomentumchange 144
Angularmomentumandtorque 149
Conservationlawofangularmomentum 150
Lawofconservationofthelinearmomentum 150
Summary 150
Thelawofareas 151
Conservationoforientation 151
CONTENTS xi
18 TheGeneralLinearMotion 159
19 TheFreeFall 163
Verticalthrow 164
Inclinedthrow 166
20 Friction 172
Frictionphenomenainaviscousmedium 172
MotioninaviscousmediumwithNewtonianfriction 177
Generalizedansatzforfriction: 179
21 TheHarmonicOscillator 196
22 MathematicalInterlude—SeriesExpansion,Euler’sFormulas 210
23 TheDampedHarmonicOscillator 214
24 ThePendulum 229
25 MathematicalInterlude:DifferentialEquations 241
26 PlanetaryMotions 246
27 SpecialProblemsinCentralFields 282
Thegravitationalfieldofextendedbodies 282
Theattractiveforceofasphericalmassshell 283
Thegravitationalpotentialofasphericalshellcoveredwithmass 285
Stabilityofcircularorbits 289
28 TheEarthandourSolarSystem 295
Generalnotionsofastronomy 295
Determinationofastronomicquantities 296
Properties,position,andevolutionofthesolarsystem 308
Worldviews 315
Ontheevolutionoftheuniverse 325
DarkMatter 330
Whatisthenatureofthedarkmatter? 338