This book is devoted to explaining a wide range of applications of continuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems.
本書為英文版。
片斷：
Thefirstsectiongivesabasicoutlineofthegeneralconceptofamanifold,
theseconddoingthesameforLiegroups,bothlocalandglobal.Inpractice
Liegroupsariseasgroupsofsymmetriesofsomeobject,or,moreprecisely,
aslocalgroupsoftransformationsactingonsomemanifold;thesecondsec-
tiongivesabrieflookatthese.Themostimportantconceptintheentire
theoryisthatofavectorfield,whichactsasthe"infmitesimalgenerator"
ofsomeone-parameterLiegroupoftransformations.Thisconceptisfun-
damentalforboththedevelopmentofthetheoryofLiegroupsandthe
applicationstodifferentialequations.Ithasthecrucialeffectofreplacing
complicatednonlinearconditionsforthesymmetryofsomeobjectundera
groupoftransformationsbyeasilyverifiablelinearconditionsreflectingits
infinitesimalsymmetryunderthecorrespondingvectorfields.Thistechnique
willbeexploredindepthforsystemsofalgebraicanddifferentialequations
inChapter2.ThenotionofvectorfieldthenleadstotheconceptofaLie
algebra,whichcanbethoughtofastheinfinitesimalgeneratoroftheLie
groupitself,thetheoryofwhichisdevelopedinSection1.4.Thefinalsection
ofthischaptergivesabriefintroductiontodifferentialformsandintegration
onmanifolds.
1.1.Manifolds
Throughoutmostofthisbook,wewillbeprimarilyinterestedinobjects,
suchasdifferentialequations,symmetrygroupsandsoon,whicharedefined
onopensubsetsofEuclideanspaceR.Theunderlyinggeometricalfeatures
oftheseobjectswillbeindependentofanyparticularcoordinatesystem
ontheopensubsetwhichmightbeusedtodefinethem,anditbecomesof
greatimportancetofreeourselvesfromthedependenceonparticularlocal
coordinates,sothatourobjectswillbeessentially"coordinate-free".More
specifically,ifURisopenand:U->V,whereV1=Risopen,isany
diffeomorphism,meaningthatisaninfinitelydifferentiablemapwithinfi-
nitelydifferentiableinverse,thenobjectsdefinedonUwillhaveequivalent
counterpartsonV.AlthoughthepreciseformulaefortheobjectonUandits
counterpartonVwill,ingeneral,change,theessentialunderlyingproperties
willremainthesame.Oncewehavefreedourselvesfromthisdependenceon
coordinates,itisasmallsteptothegeneraldefmitionofasmoothmanifold.
Fromthispointofview,manifoldsprovidethenaturalsettingforstudying
objectsthatdonotdependoncoordinates.
|
