Theoretical Physics 2: Analytical Mechanics

371 Pages·2016·2.36 MB·English

by  Wolfgang Nolting

Wolfgang Nolting
Theoretical
Physics 2
Analytical Mechanics
Theoretical Physics 2

Wolfgang Nolting
Theoretical Physics 2
Analytical Mechanics
123
WolfgangNolting
Inst.Physik
Humboldt-UniversitaRtzuBerlin
Berlin,Germany
ISBN978-3-319-40128-7 ISBN978-3-319-40129-4 (eBook)
DOI10.1007/978-3-319-40129-4
LibraryofCongressControlNumber:2016943655
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General Preface
ThesevenvolumesoftheseriesBasicCourse:TheoreticalPhysicsarethoughttobe
textbookmaterialforthestudyofuniversity-levelphysics.Theyareaimedtoimpart,
in a compact form, the most important skills of theoretical physics which can be
usedasbasisforhandlingmoresophisticatedtopicsandproblemsintheadvanced
study of physics as well as in the subsequent physics research. The conceptual
designofthepresentationisorganizedinsuchawaythat
ClassicalMechanics(volume1)
AnalyticalMechanics(volume2)
Electrodynamics(volume3)
SpecialTheoryofRelativity(volume4)
Thermodynamics(volume5)
are considered as the theory part of an integrated course of experimental and
theoretical physics as is being offered at many universities starting from the first
semester.Therefore,thepresentationisconsciouslychosentobeveryelaborateand
self-contained,sometimessurelyatthe costofcertainelegance,sothatthecourse
is suitableevenforself-study,at firstwithoutanyneedof secondaryliterature.At
anystage,nomaterialisusedwhichhasnotbeendealtwithearlierinthetext.This
holds in particular for the mathematical tools, which have been comprehensively
developed starting from the school level, of course more or less in the form of
recipes, such that right from the beginning of the study, one can solve problems
in theoretical physics. The mathematical insertions are always then plugged in
whentheybecomeindispensabletoproceedfurtherintheprogrammeoftheoretical
physics. It goes without saying that in such a context, not all the mathematical
statements can be proved and derived with absolute rigour. Instead, sometimes a
referencemustbemadetoanappropriatecourseinmathematicsortoanadvanced
textbook in mathematics. Nevertheless, I have tried for a reasonably balanced
representationsothatthemathematicaltoolsarenotonlyapplicablebutalsoappear
atleast‘plausible’.
Themathematicalinterludesareofcoursenecessaryonlyinthefirstvolumesof
this series, which incorporatemore or less the materialof a bachelorprogramme.
v
vi GeneralPreface
Inthesecondpartoftheserieswhichcomprisesthemodernaspectsoftheoretical
physics,
QuantumMechanics:Basics(volume6)
QuantumMechanics:MethodsandApplications(volume7)
StatisticalPhysics(volume8)
Many-BodyTheory(volume9),
mathematicalinsertionsarenolongernecessary.Thisispartlybecause,bythetime
onecomestothisstage,theobligatorymathematicscoursesonehastotakeinorder
to study physics would have provided the required tools. The fact that training in
theory has already started in the first semester itself permits inclusion of parts of
quantum mechanics and statistical physics in the bachelor programme itself. It is
clear that the content of the last three volumes cannot be part of an integrated
coursebutratherthesubjectmatterofpuretheorylectures.Thisholdsinparticular
forMany-BodyTheorywhichisoffered,sometimesunderdifferentnamesas,e.g.,
advancedquantummechanics,intheeighthorsosemesterofstudy.Inthispart,new
methodsandconceptsbeyondbasicstudiesareintroducedanddiscussedwhichare
developedinparticularforcorrelatedmanyparticlesystemswhichinthemeantime
havebecomeindispensablefora studentpursuingmaster’sora higherdegreeand
forbeingabletoreadcurrentresearchliterature.
In all the volumes of the series Basic Course: Theoretical Physics, numerous
exercisesareincludedtodeepentheunderstandingandtohelpcorrectlyapplythe
abstractlyacquiredknowledge.Itisobligatoryforastudenttoattemptonhisown
to adapt and apply the abstract concepts of theoretical physics to solve realistic
problems.Detailed solutionsto the exercisesare givenat the endof eachvolume.
The idea is to help a student to overcomeany difficulty at a particular step of the
solutionortocheckone’sowneffort.Importantlythesesolutionsshouldnotseduce
thestudenttofollowtheeasywayoutasasubstituteforhisowneffort.Attheend
of each biggerchapter,I have addedself-examinationquestionswhich shall serve
asaself-testandmaybeusefulwhilepreparingforexaminations.
I should not forget to thank all the people who have contributed one way or
an other to the success of the book series. The single volumes arose mainly from
lectures which I gave at the universities of Muenster, Wuerzburg, Osnabrueck,
and Berlin in Germany, Valladolid in Spain, and Warangal in India. The interest
and constructivecriticism of the students providedme the decisive motivationfor
preparing the rather extensive manuscripts. After the publication of the German
version,Ireceivedalotofsuggestionsfromnumerouscolleaguesforimprovement,
and this helped to further develop and enhance the concept and the performance
of the series. In particular, I appreciate very much the support by Prof. Dr. A.
Ramakanth,a long-standingscientific partnerand friend,who helpedme in many
respects,e.g.whatconcernsthecheckingofthetranslationoftheGermantextinto
thepresentEnglishversion.
GeneralPreface vii
SpecialthanksareduetotheSpringercompany,inparticulartoDr.Th.Schneider
and his team. I remember many useful motivations and stimulations. I have the
feelingthatmybooksarewelltakencareof.
Berlin,Germany WolfgangNolting
May2015

Preface to Volume 2
The concern of classical mechanics consists in the setting up and solving of
equationsofmotionfor
masspoints,systemofmasspoints,rigidbodies
onthebasisofasfewaspossible
axiomsandprinciples.
Thelatterare mathematicallynotstrictlyprovablebutrepresentmerelyupto now
self-consistent facts of everyday experience. One might of course ask why one
even today still deals with classical mechanics although this discipline may have
a direct relationship to current research only in very rare cases. On the other
hand,classicalmechanicsrepresentstheindispensablebasisforthemoderntrends
of theoretical physics, which means they cannot be put across without a deep
understanding of classical mechanics. Furthermore, as a side effect, mechanics
permits in connection with relatively familiar problems a certain habituation to
mathematical algorithms. So we have exercised intensively in the first volume of
thisBasicCourse:TheoreticalPhysicsinconnectionwithNewton’sMechanicsthe
inputofvectoralgebra.
Why, however,are we dealing in this second volume once more with classical
mechanics? The analytical mechanics of the underlying second volume treats
the formulations according to Lagrange, Hamilton, and Hamilton-Jacobi, which,
strictlyspeaking,donotpresentanynewphysicscomparedtotheNewtonianversion
being, however, methodically much more elegant and, what is more, revealing a
moredirectreferencetoadvancedcoursesintheoreticalphysicssuchasthequantum
mechanics.
Themaingoalofthisvolume2correspondsexactlytothatofthe totalGround
Course:TheoreticalPhysics.Itisthoughttobeanaccompanyingtextbookmaterial
forthe studyof university-levelphysics.Itis aimedto impart,in a compactform,
the most important skills of theoretical physics which can be used as basis for
handlingmoresophisticatedtopicsandproblemsintheadvancedstudyofphysics
as well as in the subsequent physics research. It is presented in such a way that
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