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. 本书为英文版。 片断： 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.
