Differential Galois Theory and Non-Integrability of Hamiltonian Systems

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Springer Science & Business Media, 1 août 1999 - 167 pages
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This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Liapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc.

The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed.

 

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Table des matières

Introduction
1
Differential Galois Theory
7
22 Classical approach
11
23 Meromorphic connections
15
24 The Tannakian approach
24
25 Stokes multipliers
25
26 Coverings and differential Galois groups
28
27 Kovacics algorithm
29
512 Nonintegrability theorem
98
513 Examples
101
52 The Bianchi IX cosmological model
105
522 Nonintegrability
107
53 Sitnikovs ThreeBody Problem
109
532 Nonintegrability
110
An Application of the Lame Equation
111
61 Computation of the potentials
112

28 Examples
33
282 The Bessel equation
34
283 The confluent hypergeometric equation
36
284 The Lame equation
37
Hamiltonian Systems
43
31 Definitions
44
32 Complete integrability
48
33 Three nonintegrability theorems
52
34 Some properties of Poisson algebras
57
Nonintegrability Theorems
65
41 Variational equations
66
412 Meromorphic connection associated with the variational equation
70
413 Reduction to normal variational equations
72
414 Reduction from the Tannakian point of view
78
42 Main results
80
43 Examples
90
Three Models
97
62 Nonintegrability criterion
115
63 Examples
123
64 The homogeneous HenonHeiles potential
126
A Connection with Chaotic Dynamics
131
71 GrottaRagazzo interpretation of Lermans theorem
132
72 Differential Galois approach
133
73 Example
135
Complementary Results and Conjectures
139
82 A conjecture about the dynamic
142
832 An application
146
Meromorphic Bundles
149
Galois Groups and Finite Coverings
153
Connections with Structure Group
157
Bibliography
159
Index
167
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À propos de l'auteur (1999)

Juan J. Morales Ruiz is Professor at the Universidad Politécnica de Madrid, Spain.

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