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Equivalence, Invariants, and Symmetry
PETER J. OLVER University of Minnesota Presented by Mehdi Nadjafikhah webpages.iust.ac.ir\m_nadjafikhah/ Last Update: January 25, 2010
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Contents Preface Acknowledgments Introduction 1. Geometric Foundations
Contents Preface Acknowledgments Introduction 1. Geometric Foundations Manifolds Functions Submanifolds Vector Fields Lie Brackets The Differential Differential Forms Equivalence of Differential Forms 29
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2. Lie Groups Transformation Groups Invariant Subsets and Equations 39 Canonical Forms Invariant Functions Lie Algebras Structure Constants The Exponential Map Subgroups and Subalgebras Infinitesimal Group Actions Classification of Group Actions 58 Infinitesimal Invariance Invariant Vector Fields Lie Derivatives and Invariant Differential Forms 68 The Maurer-Cartan Forms
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3. Representation Theory 75
Representations Representations on Function Spaces 81 Multiplier Representations linfinitesimal Multipliers Relative Invariants Classical Invariant Theory
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4. Jets and Contact Transformations 105
Transformations and Functions Invariant Functions Jets and Prolongations Total Derivatives Prolongation of Vector Fields Contact Forms Contact Transformations Infinitesimal Contact Transformations Classification of Groups of Contact Transformations134
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5. Differential Invariants 136
Dimensional Considerations Infinitesimal Methods Stabilization and Effectiveness Invariant Differential Operators Invariant Differential Forms Several Dependent Variables Several Independent Variables
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6. Symmetries of Differential Equations 175
Symmetry Groups and Differential Equations 175 Infinitesimal Methods Integration of Ordinary Differential Equations 187 Characterization of Invariant Differential Equations 191 Lie Determinants Symmetry Classification of Ordinary Differential Equations A Proof of Finite Dimensionality Linearization of Partial Differential Equations 209 Differential Operators Applications to the Geometry of Curves 218
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7. Symmetries of Variational Problems 221
The Calculus of Variations Equivalence of Functionals Invariance of the Euler-Lagrange Equations 230 Symmetries of Variational Problems Invariant Variational Problems Symmetry Classification of Variational Problems 240 First Integrals The Cartan Form Invariant Contact Forms and Evolution Equations 246
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8. Equivalence of Coframes 252
Frames and Coframes The Structure Functions Derived Invariants Classifying Functions The Classifying Manifolds Symmetries of a Coframe Remarks and Extensions
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9. Formulation of Equivalence Problems 280
Equivalence Problems Using Differential Forms 280 Coframes and Structure Groups 287 Normalization Overdetermined Equivalence Problems 297
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10. Cartan's Equivalence Method 304
The Structure Equations Absorption and Normalization Equivalence Problems for Differential Operators Fiber-preserving Equivalence of Scalar Lagrangians An Inductive Approach to Equivalence Problems Lagrangian Equivalence under Point Transformations Applications to Classical Invariant Theory 333 Second Order Variational Problems Multi-dimensional Lagrangians
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11. Involution Citrtan's Test Intransitive Case Divergence Equivalence of First Order 357 Lagrangians Intrinsic Method Contact Transformations Darboux’ Theorem Intransitive Case Equivalence of Nonclosed Two-Forms 367
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12. Prolongation of Equivalence Problems 372
The Determinate Case Equivalence of Surfaces Conformal Equivalence of Surfaces Equivalence of Riemannian Manifolds The Indeterminate Case Second Order Ordinary Differential Equations 397
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13. Differential Systems. 409. Differential Systems and Ideals. 409
13. Differential Systems Differential Systems and Ideals Equivalence of Differential Systems Vector Field Systems
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14. Frobenius' Theorem Vector Field Systems Differential Systems Characteristics and Normal Forms The Technique of the Graph Global Equivalence
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15. The Cartan-Kahler Existence Theorem 447
The Cauchy-Kovalevskaya Existence Theorem 447 Necessary Conditions Sufficient Conditions Applications to Equivalence Problems 460 Involutivity and Transversality Tables References Symbol Index Author Index Subject Index
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This book prove to be a stimulating, unusual, and provocative blend of mathematical flavors.
As its title indicates, the book revolves around three interconnected and particularly fertile themes, each arising in a wide variety of mathematical disciplines, and each having a wealth of significant and substantial applications. Equivalence deals with the determination of when two mathematical objects are the same under a change of variables. The symmetries of a given object can be interpreted as the group of self-equivalences. Conditions guaranteeing equivalence are most effectively expressed in terms of invariants, whose values are unaffected by the changes of variables.
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The book naturally divides into four interconnected parts.
The first, comprising Chapters 1-3, constitutes the algebro-geometric foundation of our subject. The Second part, comprising Chapters 4-7, provides an in depth study of applications of symmetry methods to differential equations. In the third part, Chapters 8-12, the focus shifts to equivalence problems, and the Cartan approach to their solution. The final three chapters survey the required results from the theory of partial differential equations and differential systems.
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The basic prerequisites for the book are
Multi-variable calculus (specifically the implicit and inverse function theorems and the divergence theorem) Basic tensor and exterior algebra, and a smattering of group theory Results from elementary linear algebra and complex analysis, and Basic existence theorems for ordinary differential equations are used without comment.
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It is my hope that this book will serve as a catalyst for the further development, both in theory and in applications, of this fascinating and fertile mathematic field. I am certain that there are many fundamental contributions yet to be made, and that the devoted student cannot help but play a role in its accelerating mathematical development, or in its ever-broadening range of applications.
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