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| 作者 |
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P.J.Olver
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| ISBN |
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7506207303
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| 页数 |
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513
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| 开本 |
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24开
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| 封面形式 |
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简裝本
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| 出版社 |
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世界图书出版公司
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| 出版日期 |
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1999-11-1
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| NT$ |
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846
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配送说明： 国际快递 ， 海运邮递 。
付款说明： 1. VISA、MASTER線上刷卡 2. 信用卡传真刷卡付款 3.
邮政划拨 4. 银行汇款
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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。
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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.
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Preface to First Edition
Preface to Second Edition
Acknowledgments
Introduction
Notes to the Reader
CHAPTER 1 Introduction to Lie Groups
1.1 Manifolds
Change ofCoordinates
Maps Between Manifolds
The Maximal Rank Condition
Submanifolds
Reaular Submanifolds
Implicit Submanifolds
Curves and Connectedness
1.2 Lie Groups
Lie Subgroups
Local Lie Groups
Local Transformation Groups
Orbits
1.3. Vector Fields
Flows
Action on Functions
Differentials
Lie Brackets
Tangent Spaces and Vectors Fields on Submanifolds
Frobenius' Theorem
1.4 LieAlgebras
One-Parameter Subgroups
Subalgebras
The Exponential Map
Lie Algebras of Local Lie Groups
Structure Constants
Commutator Tables
Infinitesimal Group Actions
1.5. Differential Forms
PuIl-Back and Change ofCoordinates
Interior Products
The Differential
The de Rham Complex
Lie Derivatives
Homotopy Operators
Integration and Stokes' Theorem
Notes
Exercises
CHAPTER 2 Symmetry Groups of DifTerential Equations
2.1 Symmetries of Algebraic Equations
Invariant Subsets
Invariant Functions
Infinitesimal Invariance
Local Invariance
Invariants and Functional Dependence
Methods for Constructing Invariants
2.2. Groups and Differential Equations
2.3. Prolongation
Systems of DifTerential Equations
Prolongation ofGroup Actions
Invariance of Differential Equations
Prolongation of Vector Fields
Infinitesimal Invariance
The Prolongation Formula
Total Derivatives
The General Prolongation Formula
Properties of Prolonged Vector Fields
Characteristics of Symmetries
2.4. Calculation of Symmetry Groups
2.5. Integration of Ordinary Differential Equations
First Order Equations
Higher Order Equations
Differential Invariants
Multi-parameter Symmetry Groups
Solvable Groups
Systems of Ordinary Differential Equations
2.6. Nondegeneracy Conditions for Differential Equations
Local Solvability
Invariance Criteria
The Cauchy-Kovalevskaya Theorem
Characteristics
Normal Systems
Prolongation of DifTerential Equations
Notes
Exercises
CHAPTER 3 Group-Invariant Solutions
……
CHAPTER 4 Symmetry Groups and Conservation Laws
CHAPTER 5 Generalized Symmetries
CHAPTER 6 Finite-Dimensional Hamiltonian Systems
CHAPTER 7 Hamiltonian Methods for Evolution Equations
References
Symbol Index
Author Index
Subject Index
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