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Morales-Ruiz J.J. — Differential Galois Theory and Non-Integrability of Hamiltonian Systems
Morales-Ruiz J.J. — Differential Galois Theory and Non-Integrability of Hamiltonian Systems

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Название: Differential Galois Theory and Non-Integrability of Hamiltonian Systems

Автор: Morales-Ruiz J.J.


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.

Язык: en

Рубрика: Математика/Алгебра/Дифференциальная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 183

Добавлена в каталог: 18.03.2005

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Предметный указатель
Algebraic normal variational equation (ANVE)      103
Baldassarri      41
Bessel equation      37
Bianchi IX cosmologieal model      109
Brioschi determinant      45
Differential Held      13
Dwork      43
Exact sequence of connections      23
Functor fibre      26
Galois correspondence      13 14
Galois group      13
Grauert theorem      162
Grotta — Ragazzo      140
Halphen      43
Hamiltonian system      49
Henon — Heiles Hamiltonians      108 130 132 157
Higher order variational equations      152
Holomorphic connection      17
Homoclinic orbit      60
Homogeneous potentials      101
Hypergeometric equation      35
Identity component      10
Integrability      14 52
Ito      131
Katz      168
Kimura's theorem      36
Kolchin      13 169
Kovacic's algorithm      31
Lame equation      39
Lermairs theorem      60
Liapounov orbits      60
Lie — Kolchin theorem      10
Linear algebraic group      9
Liouvilkrs theorem      54
Marotte      167
Meromorphic connection      19
Meromorphic vector bundle      161
Monodromy group      14
Normal variational equation      58 76
Painleve test      110 159
Picard — Vessiot extension      13
Poincare — Arnold — Mellnikov theory      159
Poincare's theorem      57
Poisson algebra      50
Ramis density theorem      29
Riccati equation      15
Schwarz      36
Serre      163
Sitnikov throe body problem      114
Spring Pendulum system      149
Stokes multipliers      27
Symplectic bundle      24
Symplectic connection      25
Symploctic manifold      48
Tannakian      26
Taub solutions      111
Toda lattice      132
Variational equation (VE)      57 70
Whittaker equation      138
Youshida theorem      104
Zariski topology      10
Ziglin's lemma      90
Ziglin's theorem      59
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