Algebraic integrability, Painlevé geometry and Lie algebras:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; 47 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 483 S. graph. Darst. |
| ISBN: | 354022470X |
Staff View
MARC
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| 100 | 1 | |a Adler, Mark |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic integrability, Painlevé geometry and Lie algebras |c Mark Adler ; Pierre van Moerbeke ; Pol Vanhaecke |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XII, 483 S. |b graph. Darst. | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v 47 | |
| 650 | 4 | |a Integrables System - Algebraische Geometrie - Lie-Theorie | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Geometry, Algebraic | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Painlevé equations | |
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Record in the Search Index
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| adam_text | Table of Contents
1 Introduction 1
Part I Liouville Integrable Systems
2 Lie Algebras 7
2.1 Structures on Manifolds 7
2.1.1 Vector Fields and 1 Forms 7
2.1.2 Distributions and the Frobenius Theorem 11
2.1.3 Differential Forms and Poly vector Fields 13
2.1.4 Lie Derivatives 15
2.2 Lie Groups and Lie Algebras 16
2.3 Simple Lie Algebras 22
2.3.1 The Classification 22
2.3.2 Invariant Functions and Exponents 29
2.4 Twisted Affine Lie Algebras 33
3 Poisson Manifolds 41
3.1 Basic Definitions 43
3.2 Hamiltonian Mechanics 47
3.3 Bi Hamiltonian Manifolds and Vector Fields 53
3.4 Local and Global Structure 55
3.5 The Lie Poisson Structure of g* 57
3.6 Constructing New Poisson Manifolds from Old Ones 62
4 Integrable Systems on Poisson Manifolds 67
4.1 Functions in Involution 67
4.2 Liouville Integrability 73
4.3 The Liouville Theorem and the Action Angle Theorem 78
4.4 The Adler Kostant Symes Theorem(s) 82
4.4.1 Lie Algebra Splitting 83
4.4.2 The AKS Theorem on g* 84
4.4.3 .R Brackets and Double Lie Algebras 88
4.4.4 The AKS Theorem on g 89
4.5 Lax Operators and r matrices 96
X Table of Contents
Part II Algebraic Completely Integrable Systems
5 The Geometry of Abelian Varieties 107
5.1 Algebraic Varieties versus Complex Manifolds 107
5.1.1 Notations and Terminology 107
5.1.2 Divisors and Line Bundles 108
5.1.3 Projective Embeddings of Complex Manifolds 113
5.1.4 Riemann Surfaces and Algebraic Curves 117
5.2 Abelian Varieties 121
5.2.1 The Riemann Conditions 122
5.2.2 Line Bundles on Abelian Varieties and Theta Functions... 125
5.2.3 Jacobian Varieties 129
5.2.4 Prym Varieties 135
5.2.5 Families of Abelian Varieties 139
5.3 Divisors in Abelian Varieties 141
5.3.1 The Case of Non singular Divisors 143
5.3.2 The Case of Singular Divisors 146
6 A.c.i. Systems 153
6.1 Definitions and First Examples 154
6.2 Necessary Conditions for Algebraic Complete Integrability 164
6.2.1 The Kowalevski Painleve Criterion 164
6.2.2 The Lyapunov Criterion 176
6.3 The Complex Liouville Theorem 180
6.4 Lax Equations with a Parameter 184
7 Weight Homogeneous A.c.i. Systems 199
7.1 Weight Homogeneous Vector Fields and Laurent Solutions 200
7.2 Convergence of the Balances 213
7.3 Weight Homogeneous Constants of Motion 215
7.4 The Kowalevski Matrix and its Spectrum 218
7.5 Weight Homogeneous A.c.i. Systems 226
7.6 Algorithms 229
7.6.1 The Indicial Locus 1 and the Kowalevski Matrix /C 229
7.6.2 The Principal Balances (for all Vector Fields) 230
7.6.3 The Constants of Motion 234
7.6.4 The Abstract Painleve Divisors Tc 236
7.6.5 Embedding the Tori Trc 238
7.6.6 The Quadratic Differential Equations 240
7.6.7 The Holomorphic Differentials on T c 242
7.7 Proving Algebraic Complete Integrability 245
7.7.1 Embedding the Tori T^ and Adjunction 247
7.7.2 Extending One of the Vector Fields XF 252
7.7.3 Going into the Affine 254
Table of Contents XI
Part III Examples
8 Integrable Geodesic Flow on SO(4) 265
8.1 Geodesic Flow on SO(4) 265
8.1.1 From Geodesic Flow on G to a Hamiltonian Flow on g ... 265
8.1.2 Half diagonal Metrics on so(4) 267
8.1.3 The Kowalevski Painleve Criterion 270
8.2 Geodesic Flow for the Manakov Metric 289
8.2.1 From Metric I to the Manakov Metric 289
8.2.2 A Curve of Rank Three Quadrics 293
8.2.3 A Normal Form for the Manakov Metric 295
8.2.4 Algebraic Complete Integrability of the Manakov Metric .. 297
8.2.5 The Invariant Manifolds as Prym Varieties 308
8.2.6 A.c.i. Diagonal Metrics on so(4) 315
8.2.7 From the Manakov Flow to the Clebsch Flow 318
8.3 Geodesic Flow for Metric II and Hyperelliptic Jacobians 321
8.3.1 A Normal Form for Metric II 321
8.3.2 Algebraic Complete Integrability 325
8.3.3 A Lax Equation for Metric II 334
8.3.4 From Metric II to the Lyapunov Steklov Flow 337
8.4 Geodesic Flow for Metric III and Abelian Surfaces of Type (1,6) 339
8.4.1 A Normal Form for Metric III 339
8.4.2 A Lax Equation for Metric III 342
8.4.3 Algebraic Complete Integrability 344
9 Periodic Toda Lattices Associated to Cartan Matrices 361
9.1 Different Forms of the Periodic Toda Lattice 361
9.2 The Kowalevski Painleve Criterion 365
9.3 A Lax Equation for the Periodic Toda Lattice 371
9.4 Algebraic Integrability of the o^ Toda Lattice 376
9.5 The Geometry of the Periodic Toda Lattices 386
9.5.1 Notation 386
9.5.2 The Balances of the Periodic Toda Lattice 389
9.5.3 Equivalence of Painleve Divisors 394
9.5.4 Behavior of the Principal Balances Near the Lower Ones .. 398
9.5.5 Tangency of the Toda Flows to the Painleve Divisors 403
9.5.6 Intersection Multiplicity of Two Painleve Divisors 409
9.5.7 Toda Lattices Leading to Abelian Surfaces 412
9.5.8 Intersection Multiplicity of Many Painleve Divisors 416
XII Table of Contents
10 Integrable Spinning Tops 419
10.1 Spinning Tops 419
10.1.1 Equations of Motion and Poisson Structure 419
10.1.2 A.c.i. Tops 424
10.2 The Euler Poinsot and Lagrange Tops 428
10.2.1 The Euler Poinsot Top 428
10.2.2 The Lagrange Top 433
10.3 The Kowalevski Top 436
10.3.1 Liouville Integrability and Lax Equation 436
10.3.2 Algebraic Complete Integrability 443
10.4 The Goryachev Chaplygin Top 453
10.4.1 Liouville Integrability and Lax Equation 453
10.4.2 The Bechlivanidis van Moerbeke System 455
10.4.3 Almost Algebraic Complete Integrability 465
10.4.4 The Relation Between the Toda and the Bechlivanidis
van Moerbeke System 466
References 469
Index 479
|
| any_adam_object | 1 |
| author | Adler, Mark Van Moerbeke, Pierre Vanhaecke, Pol 1963- |
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| dewey-ones | 512 - Algebra |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:59:17Z |
| institution | BVB |
| isbn | 354022470X |
| language | English |
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| physical | XII, 483 S. graph. Darst. |
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| series | Ergebnisse der Mathematik und ihrer Grenzgebiete |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spelling | Adler, Mark Verfasser aut Algebraic integrability, Painlevé geometry and Lie algebras Mark Adler ; Pierre van Moerbeke ; Pol Vanhaecke Berlin [u.a.] Springer 2004 XII, 483 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge 47 Integrables System - Algebraische Geometrie - Lie-Theorie Differential equations Geometry, Algebraic Lie algebras Painlevé equations Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Lie-Theorie (DE-588)4251836-2 gnd rswk-swf Integrables System (DE-588)4114032-1 gnd rswk-swf Integrables System (DE-588)4114032-1 s Algebraische Geometrie (DE-588)4001161-6 s Lie-Theorie (DE-588)4251836-2 s DE-604 Van Moerbeke, Pierre Verfasser aut Vanhaecke, Pol 1963- Verfasser (DE-588)1028305303 aut Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge ; 47 (DE-604)BV000899194 47 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012857015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Adler, Mark Van Moerbeke, Pierre Vanhaecke, Pol 1963- Algebraic integrability, Painlevé geometry and Lie algebras Ergebnisse der Mathematik und ihrer Grenzgebiete Integrables System - Algebraische Geometrie - Lie-Theorie Differential equations Geometry, Algebraic Lie algebras Painlevé equations Algebraische Geometrie (DE-588)4001161-6 gnd Lie-Theorie (DE-588)4251836-2 gnd Integrables System (DE-588)4114032-1 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4251836-2 (DE-588)4114032-1 |
| title | Algebraic integrability, Painlevé geometry and Lie algebras |
| title_auth | Algebraic integrability, Painlevé geometry and Lie algebras |
| title_exact_search | Algebraic integrability, Painlevé geometry and Lie algebras |
| title_full | Algebraic integrability, Painlevé geometry and Lie algebras Mark Adler ; Pierre van Moerbeke ; Pol Vanhaecke |
| title_fullStr | Algebraic integrability, Painlevé geometry and Lie algebras Mark Adler ; Pierre van Moerbeke ; Pol Vanhaecke |
| title_full_unstemmed | Algebraic integrability, Painlevé geometry and Lie algebras Mark Adler ; Pierre van Moerbeke ; Pol Vanhaecke |
| title_short | Algebraic integrability, Painlevé geometry and Lie algebras |
| title_sort | algebraic integrability painleve geometry and lie algebras |
| topic | Integrables System - Algebraische Geometrie - Lie-Theorie Differential equations Geometry, Algebraic Lie algebras Painlevé equations Algebraische Geometrie (DE-588)4001161-6 gnd Lie-Theorie (DE-588)4251836-2 gnd Integrables System (DE-588)4114032-1 gnd |
| topic_facet | Integrables System - Algebraische Geometrie - Lie-Theorie Differential equations Geometry, Algebraic Lie algebras Painlevé equations Algebraische Geometrie Lie-Theorie Integrables System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012857015&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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