Differential Galois Theory and Non-Integrability of Hamiltonian Systems

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

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 withi...

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Bibliographic Details
Main Author: Morales Ruiz, Juan J. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Basel : Springer Basel : Imprint: Birkhäuser, 1999.
Edition:1st ed. 1999.
Series:Progress in Mathematics, 179
Springer eBook Collection.
Subjects:
Differential equations.
Global analysis (Mathematics).
Manifolds (Mathematics).
Algebra.
Field theory (Physics).
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 Introduction
  • 2 Differential Galois Theory
  • 2.1 Algebraic groups
  • 2.2 Classical approach
  • 2.3 Meromorphic connections
  • 2.4 The Tannakian approach
  • 2.5 Stokes multipliers
  • 2.6 Coverings and differential Galois groups
  • 2.7 Kovacic’s algorithm
  • 2.8 Examples
  • 3 Hamiltonian Systems
  • 3.1 Definitions
  • 3.2 Complete integrability
  • 3.3 Three non-integrability theorems
  • 3.4 Some properties of Poisson algebras
  • 4 Non-integrability Theorems
  • 4.1 Variational equations
  • 4.2 Main results
  • 4.3 Examples
  • 5 Three Models
  • 5.1 Homogeneous potentials
  • 5.2 The Bianchi IX cosmological model
  • 5.3 Sitnikov’s Three-Body Problem
  • 6 An Application of the Lamé Equation
  • 6.1 Computation of the potentials
  • 6.2 Non-integrability criterion
  • 6.3 Examples
  • 6.4 The homogeneous Hénon-Heiles potential
  • 7 A Connection with Chaotic Dynamics
  • 7.1 Grotta-Ragazzo interpretation of Lerman’s theorem
  • 7.2 Differential Galois approach
  • 7.3 Example
  • 8 Complementary Results and Conjectures
  • 8.1 Two additional applications
  • 8.2 A conjecture about the dynamic
  • 8.3 Higher-order variational equations
  • A Meromorphic Bundles
  • B Galois Groups and Finite Coverings
  • C Connections with Structure Group.

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