Overview

This monograph is devoted to the relation between two different concepts of integrability leading to algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered generalizationsof classical non-integrability results by Poincare and Liapunov, as well as more recent results by Ziglin and Yoshida. Several non-mathematical applications are given.

Full Product Details

Author:   Juan J. Morales Ruiz
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1999 ed.
Volume:   v. 179
Dimensions:   Width: 15.60cm , Height: 1.20cm , Length: 23.40cm
Weight:   0.438kg
ISBN:  

9783764360788

ISBN 10:   376436078
Pages:   181
Publication Date:   01 August 1999
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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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.- 2.8.1 The hypergeometric equation.- 2.8.2 The Bessel equation.- 2.8.3 The confluent hypergeometric equation.- 2.8.4 The Lame equation.- 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.- 4.1.1 Singular curves.- 4.1.2 Meromorphic connection associated with the variational equation.- 4.1.3 Reduction to normal variational equations.- 4.1.4 Reduction from the Tannakian point of view.- 4.2 Main results.- 4.3 Examples.- 5 Three Models.- 5.1 Homogeneous potentials.- 5.1.1 The model.- 5.1.2 Non-integrability theorem.- 5.1.3 Examples.- 5.2 The Bianchi IX cosmological model.- 5.2.1 The model.- 5.2.2 Non-integrability.- 5.3 Sitnikov's Three-Body Problem.- 5.3.1 The model.- 5.3.2 Non-integrability.- 6 An Application of the Lame Equation.- 6.1 Computation of the potentials.- 6.2 Non-integrability criterion.- 6.3 Examples.- 6.4 The homogeneous Henon-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.- 8.3.1 A conjecture.- 8.3.2 An application.- A Meromorphic Bundles.- B Galois Groups and Finite Coverings.- C Connections with Structure Group.

Reviews

...[an] account of recent work of the author and co-workers on obstructions to the complete integrability of complex Hamiltonian systems. The methods are of considerable importance to practitioners... The book provides all the needed background...and presents concrete examples in considerable detail... The final chapter...includes a fascinating account of work-in-progress by the author and his collaborators... Of particular interest...is the program of extending the differential Galois theory to higher-order variational equations... [an] excellent introduction to non-integrability methods in Hamiltonian mechanics [that] brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. --Mathematical Reviews

.,. [an] account of recent work of the author and co-workers on obstructions to the complete integrability of complex Hamiltonian systems. The methods are of considerable importance to practitioners... The book provides all the needed background...and presents concrete examples in considerable detail... The final chapter...includes a fascinating account of work-in-progress by the author and his collaborators... Of particular interest...is the program of extending the differential Galois theory to higher-order variational equations... [an] excellent introduction to non-integrability methods in Hamiltonian mechanics [that] brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. <p>--Mathematical Reviews

Author Information

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

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