The lazy universe. An introduction to the principle of least action

281 Pages·2017·4.7 MB·English

by  Coopersmith

THE LAZY UNIVERSE
FromthearchivesoftheDublinInstituteforAdvancedStudies
THE LAZY
UNIVERSE
An Introduction to the Principle
of Least Action
by
Jennifer Coopersmith
3
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof
OxfordUniversityPressintheUKandincertainothercountries
©JenniferCoopersmith2017
Themoralrightsoftheauthorhavebeenasserted
FirstEditionpublishedin2017
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin
aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe
priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted
bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics
rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe
aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe
addressabove
Youmustnotcirculatethisworkinanyotherform
andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress
198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressControlNumber:2016953473
ISBN978–0–19–874304–0
Printedandboundby
CPIGroup(UK)Ltd,Croydon,CR04YY
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand
forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials
containedinanythirdpartywebsitereferencedinthiswork.
To
CorneliusLanczos(1893–1974)
and
MurrayJohnPeake(1952–)
“There’shusbandryinheaven”
ACT II, SCENE I MACBETH
SHAKESPEARE
Preface
Themathematicianandphysicist,CorneliusLanczos(1893–1974),wrote
one of the best books of physics explanation that has ever been writ-
ten:TheVariationalPrinciplesofMechanics.1Itexplainsthetruemeaningand
philosophicalcontentofthePrincipleofLeastAction.Thepresentbook
is a shorter and simplified version of that classic - but it is an original
interpretation(anythingtakendirectlyfromLanczosisattributedand
referencedbypagenumber);itisalsoapaeantoLanczos.
Whoisthebookfor?Itisforyou!Itconcernsaprinciplethatunderpins
thewholeofphysics,sohowcoulditnotbeimportanttounderstand
evenatinybitofit?Thegreatesttheorieshaveonecommonfeature-
theyalwaysbringinwisdomfar-surpassingtheiroriginalremit,extend-
ingintoalmosteverywalkoflife.Therefore,oneshouldnotlimitone’s
areaofstudybutratherfollowthemaxim:“Themoreyouknow,the
moreyoucanknow;themoreyouunderstand,themoreyoucanunder-
stand.”2 Moreover,aswellasanincreasedunderstandingoftheworld,
youwillbeprivytoararereward-to“theoriesofexcessivebeauty”.3
It is true that the reader is assumed to have a background in the
physical sciences, however a layreader could also read this book, with
profit and enjoyment, by skim-reading the mathematics, or by read-
ingjusttheIntroduction(Chapter1),theFinalWords(Chapter9),and
the historical and popular chapters (2 and 8). Furthermore, the book
isnotatextbook: therearemanyequations,buttheaimisto explain-
whytheseequationsandnotothers,andwhatdotheequationsmean?4
Theappendicesareusuallyatamoreadvancedlevelandcondensedin
style, but they may be entirely passed over without loss of continuity
inthemaintext.Apartfromtheinvaluableassetofseeinghowaprob-
lemissolved,thereasonforsomanyappendicesistwofold:toprovidea
1 LanczosC,TheVariationalPrinciplesofMechanics,UniversityofTorontoPress(1949).All
pagenumberswillrefertothefourthedition,DoverPublications,Inc.NewYork(1970),
andweshallwrite‘Lanczos,pagex’.
2 (author’smaxim)
3 Lanczos,page229.
4 ThisisincontrasttotheapproachinSyngeandGriffith,PrinciplesofMechanics,
3rdedition,McGraw-HillBookCompany,Inc(1959),page413,wheretheadviceis:“Do
notattempttoseeaphysicalmeaninginthese[mathematical]operations;itwillnot
help.”
viii Preface
compactresourceforthephysicist(forexample,aphysicistonabeach
holiday);andforthelayreadertoknowwhatsubject-headingstofollow
upatalaterstage,ifsodesired.Enrichmentmaterialhasbeenincluded
butoftenasoptionalreading(insmallfont),orinparentheses,orfoot-
notes.Also,wehaveavoideddetailingallthequalifications,exemptions,
andspecialcases,inordertobeabletosaythingssimply,andmakebold
statements.
Terminology: on the authority of both Lanczos, and Richard
Feynman,5 we refer to all the appropriate principles - Hamilton’s
Principle, Jacobi’s Principle, Lagrange’s Principle, and Maupertuis’s
Principle - as Principles of Least Action, as they are generically so.
‘Variational Mechanics’ means any physics problem that uses the
Principle of Least Action. Also, the fact that we usually say ‘least’ as
opposedto‘stationary’isexplainedinthemaintext-seeSection6.6.
The University of La Trobe is thanked for granting me an hon-
orary research associateship, and for the use of library and comput-
ing facilities. It was Joe Petrolito, Emeritus Professor of Engineering
at La Trobe, Bendigo campus, who introduced me to the website of
EdwinFTaylor,SeniorResearchScientistEmeritusattheMassachusetts
InstituteofTechnology(MIT).Thisledtoashortbutrewardingemail
correspondence.
AtOUP,IthankSönkeAdlungandAniaWronski.TheuseofLaTeX,
Ubuntu, Gimp, Wikipedia, Google, Metapost, Scilab, and Mfpic are
gratefullyacknowledged.MalHaysomandDeborahPeakearethanked
fortheirhelpwiththediagrams(Deborahdrewthemaze,theencum-
bered porcupine and Stevin’s ‘Wreath of Spheres’). George Rogers,
librarianatDIAS,onceagainanticipatedmyneedsandsuppliedsuperb
imagesofLanczosandofHamilton,withnofuss.Mostofall,Iamgrate-
fultoGeraldSussman,PanasonicProfessorofElectricalEngineeringat
MIT, who undertook a critical reading of the book prior to printing.
The errors he identified have been corrected. Finally, I thank Murray
Peakeforhelpfuldiscussionsonmathsandphysics,andforleadingme
toLanczosinthefirstplace.
If I have inspired any reader to seek out the work of the master,
CorneliusLanczos,thenitwillbe‘missionaccomplished’.
5 FeynmanRP,Feynman’sLecturesonPhysics,VolumeII,Chapter19.
Contents
ListofFigures xi
1 Introduction 1
2 Antecedents 12
3 Mathematicsandphysicspreliminaries:ofhills
andplainsandotherthings 31
4 ThePrincipleofVirtualWork 59
5 D’Alembert’sPrinciple 88
6 LagrangianMechanics 107
7 HamiltonianMechanics 143
8 Thewholeofphysics 183
9 Finalwords 192
A1.1 Newton’sLawsofMotion 197
A2.1 Portraitsofthephysicists 198
A3.1 Reversibledisplacements 201
A6.1 WorkedexamplesinLagrangianMechanics 202
A6.2 ProofthatTisafunctionofv2 217
A6.3 Energyconservationandthehomogeneityof
time 222
A6.4 ThemethodofLagrangeMultipliers 225
A6.5 GeneralizedForces 228
A7.1 Hamilton’sTransformation,examples 230
A7.2 Demonstrationthatthep sareindependent
i
coordinates 232
A7.3 WorkedexamplesinHamiltonianMechanics 233
A7.4 Incompressibilityofthephasefluid 237
A7.5 Energyconservationinextendedphasespace 238
A7.6 Linkbetweentheaction,S,andthe‘circulation’ 241