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El. knyga: Geometry and Dynamics of Integrable Systems

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This book contains an elaborated version of the lecture notes given at the Advanced Course on Geometry andDynamics of Integrable Systems, held at the CRM in Barcelona.

Native to actualproblem-solving problems in mechanics, the topic of Integrable Systems iscurrently on the crossroad of different disciplines in pure and appliedmathematics, and it has important interactions with physics. The study of integrable systems has had special impactand also activelyuses methods of Differential Geometry. It isextremely important in SymplecticGeometry and Hamiltonian Dynamics, and has strong correlations with MathematicalPhysics, Lie Theory and Algebraic Geometry (including Mirror Symmetry). Therefore,these notes will attract experts from different backgrounds.

These notes concentrate on three different aspects of integrablesystems: obstructions to integrabilitycoming from Differential Galois theory, descriptionof singularities of integrable systems using their relation to bi-Hamiltonian systems, and generalization ofintegrable systems to the non-Hamiltonian settings. The three parts are writtenby top experts in these fields.

Foreword v
1 Integrable Systems and Differential Galois Theory
1(34)
Juan J. Morales-Ruiz
1.1 Differential Galois Theory
1(17)
1.1.1 Algebraic Groups
2(3)
1.1.2 Picard--Vessiot Theory
5(5)
1.1.3 Kovacic Algorithm
10(4)
1.1.4 Algebrization
14(2)
1.1.5 Bessel, Whittaker, and Hypergeometric Equations
16(2)
1.2 Non-Integrability of Hamiltonian Systems
18(5)
1.2.1 Introduction
18(2)
1.2.2 Homogeneous Potentials
20(3)
1.3 Integrability of the Schrodinger Equation
23(4)
1.3.1 The Schrodinger Equation
24(1)
1.3.2 Application of the Picard--Vessiot Theory
24(3)
1.4 Integrability of Fields on the Plane
27(8)
1.4.1 Riccati Fields
27(1)
1.4.2 Quadratic Polynomial Fields
28(3)
Bibliography
31(4)
2 Singularities of bi-Hamiltonian Systems and Stability Analysis
35(50)
Alexey Bolsinov
2.1 Integrable Systems: Singularities and Bifurcations
35(9)
2.1.1 Integrable Systems, Lagrangian Fibrations and their Singularities
35(4)
2.1.2 Stability and Singularities for Integrable Systems
39(1)
2.1.3 Non-Degenerate Singularities
40(4)
2.2 Jordan--Kronecker Decomposition
44(13)
2.2.1 Basic Poisson Geometry
44(3)
2.2.2 Jordan--Kronecker Decomposition
47(3)
2.2.3 Dictionary
50(1)
2.2.4 Compatible Poisson Brackets and Commuting Casimirs
51(3)
2.2.5 Properties of FΠ
54(3)
2.3 Linearisation of Poisson Pencils and a Criterion of Non-Degeneracy
57(11)
2.3.1 Linearisation of a Poisson Structure
58(1)
2.3.2 Linearisation of a Poisson Pencil
59(1)
2.3.3 Linear Pencils
60(1)
2.3.4 Non-Degenerate Linear Pencils
61(3)
2.3.5 Classification of Non-Degenerate Linear Pencils
64(2)
2.3.6 General Non-Degeneracy Criterion
66(2)
2.4 How Does It Work? Examples and Applications
68(17)
2.4.1 Rubanovskii Case
68(2)
2.4.2 Mischenko--Fomenko Systems on Semisimple Lie Algebras
70(3)
2.4.3 Euler--Manakov Tops on so(n)
73(3)
2.4.4 Periodic Toda Lattice
76(3)
Bibliography
79(6)
3 Geometry of Integrable non-Hamiltonian Systems
85
Nguyen Tien Zung
3.1 Introduction
85(3)
3.2 Normal forms, action-angle variables, and associated torus actions
88(21)
3.2.1 Liouville torus actions and action-angle variables
88(5)
3.2.2 Local normal forms of singular points
93(6)
3.2.3 Geometric linearization of non-degenerate singular points
99(8)
3.2.4 Semi-local torus actions and normal forms
107(2)
3.3 Geometry of integrable systems of type (n, 0)
109
3.3.1 Normal forms and automorphism groups
110(4)
3.3.2 Induced torus action and reduction
114(4)
3.3.3 Systems of toric degree n -- 1 and n -- 2
118(7)
3.3.4 Monodromy
125(3)
3.3.5 Totally hyperbolic actions
128(5)
3.3.6 Elbolic actions and toric manifolds
133(2)
Bibliography
135
Juan J. Morales-Ruiz is Professor of Mathematics at Universidad Politécnica de Madrid.

Alexey Bolsinov is Reader in Mathematics at Loughborough University in Leicestershire.

Nguyen Tien Zung is Professor of Mathematics at University of Toulouse.
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