Catálogo de la biblioteca Antonio Monteiro - INMABB

Includes bibliographical references (p. [469]-478) and index.

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 0* [84] --
4.4.3 R-Brackets and Double Lie Algebras [88] --
4.4.4 The AKS Theorem on 0 [89] --
4.5 Lax Operators and r-matrices [96] --
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-Painlevé 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 I and the Kowalevski Matrix K, [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 Painlevé Divisors [236] --
7.6.5 Embedding the Tori T [238] --
7.6.6 The Quadratic Differential Equations [240] --
7.6.7 The Holomorphic Differentials on D [242] --
7.7 Proving Algebraic Complete Integrability [245] --
7.7.1 Embedding the Tori Tand Adjunction [247] --
7.7.2 Extending One of the Vector Fields [252] --
7.7.3 Going into the Affine 25410 Integrable Spinning Tbps [419] --
Part III Examples --
8 Integrable Geodesic Flow on SO(4) [265] --
8.1 Geodesic Flow on SO(4) [265] --
8.1.1 FYom 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-Painlev6 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 Hyper elliptic 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-Painlevé Criterion [365] --
9.3 A Lax Equation for the Periodic Toda Lattice [371] --
9.4 Algebraic Integrability of the 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 Painlevé 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 Painlev Divisors [403] --
9.5.6 Intersection Multiplicity of Two Painlevé Divisors [409] --
9.5.7 Toda Lattices Leading to Abelian Surfaces [412] --
9.5.8 Intersection Multiplicity of Many Painlevé Divisors [416] --
10.1 Spinning Tbps --
10.1.1 Equations of Motion and Poisson Structure [419] --
10.1.2 A.c.i. Tbps [424] --
10.2 The Euler-Poinsot and Lagrange Tbps [428] --
10.2.1 The Euler-Poinsot Tbp [428] --
10.2.2 The Lagrange Tbp [433] --
10.3 The Kowalevski Tbp [436] --
10.3.1 Liouville Integrability and Lax Equation [435] --
10.3.2 Algebraic Complete Integrability [443] --
10.4 The Goryachev-Chaplygin Tbp [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 [455] --
10.4.4 The Relation Between the Toda and the Bechlivanidis-van Moerbeke System [466] --
References [469] --
lndex [479] -

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