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The Resource Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz
Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz
Resource Information
The item Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.
Resource Information
The item Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.
This item is available to borrow from all library branches.
 Summary
 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 nonintegrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical nonintegrability results by Poincaré and Lyapunov, 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, threebody problem, HénonHeiles 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.    The book is an excellent introduction to nonintegrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of workedout examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)
 Language

 eng
 eng
 Extent
 1 online resource (XIV, 167 p. 5 illus.)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 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 nonintegrability theorems
 3.4 Some properties of Poisson algebras
 4 Nonintegrability 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 ThreeBody Problem
 6 An Application of the Lamé Equation
 6.1 Computation of the potentials
 6.2 Nonintegrability criterion
 6.3 Examples
 6.4 The homogeneous HénonHeiles potential
 7 A Connection with Chaotic Dynamics
 7.1 GrottaRagazzo 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 Higherorder variational equations
 A Meromorphic Bundles
 B Galois Groups and Finite Coverings
 C Connections with Structure Group
 Isbn
 9783034887182
 Label
 Differential galois theory and nonintegrability of hamiltonian systems
 Title
 Differential galois theory and nonintegrability of hamiltonian systems
 Statement of responsibility
 Juan J. Morales Ruiz
 Language

 eng
 eng
 Summary
 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 nonintegrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical nonintegrability results by Poincaré and Lyapunov, 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, threebody problem, HénonHeiles 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.    The book is an excellent introduction to nonintegrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of workedout examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)
 Cataloging source
 MiAaPQ
 http://library.link/vocab/creatorName
 Ruiz Morales, Juan J.
 Dewey number
 515.352
 Image bit depth
 0
 Index
 index present
 Language note
 English
 LC call number
 QC173.7
 LC item number
 .R859 1999
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Progress in Mathematics
 Series volume
 Volume 179
 http://library.link/vocab/subjectName

 Field theory (Physics)
 Global analysis
 Differential equations
 Label
 Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Color
 not applicable
 Content category
 text
 Content type code
 txt
 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 nonintegrability theorems  3.4 Some properties of Poisson algebras  4 Nonintegrability 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 ThreeBody Problem  6 An Application of the Lamé Equation  6.1 Computation of the potentials  6.2 Nonintegrability criterion  6.3 Examples  6.4 The homogeneous HénonHeiles potential  7 A Connection with Chaotic Dynamics  7.1 GrottaRagazzo 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 Higherorder variational equations  A Meromorphic Bundles  B Galois Groups and Finite Coverings  C Connections with Structure Group
 Dimensions
 unknown
 Extent
 1 online resource (XIV, 167 p. 5 illus.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9783034887182
 Level of compression
 uncompressed
 Media category
 computer
 Media type code
 c
 Other control number
 10.1007/9783034887182
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000101689
 (SSID)ssj0001296419
 (PQKBManifestationID)11886452
 (PQKBTitleCode)TC0001296419
 (PQKBWorkID)11347979
 (PQKB)10865808
 (DEHe213)9783034887182
 (MiAaPQ)EBC3087619
 (EXLCZ)993400000000101689
 Label
 Differential galois theory and nonintegrability of hamiltonian systems, Juan J. Morales Ruiz
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Color
 not applicable
 Content category
 text
 Content type code
 txt
 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 nonintegrability theorems  3.4 Some properties of Poisson algebras  4 Nonintegrability 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 ThreeBody Problem  6 An Application of the Lamé Equation  6.1 Computation of the potentials  6.2 Nonintegrability criterion  6.3 Examples  6.4 The homogeneous HénonHeiles potential  7 A Connection with Chaotic Dynamics  7.1 GrottaRagazzo 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 Higherorder variational equations  A Meromorphic Bundles  B Galois Groups and Finite Coverings  C Connections with Structure Group
 Dimensions
 unknown
 Extent
 1 online resource (XIV, 167 p. 5 illus.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9783034887182
 Level of compression
 uncompressed
 Media category
 computer
 Media type code
 c
 Other control number
 10.1007/9783034887182
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000101689
 (SSID)ssj0001296419
 (PQKBManifestationID)11886452
 (PQKBTitleCode)TC0001296419
 (PQKBWorkID)11347979
 (PQKB)10865808
 (DEHe213)9783034887182
 (MiAaPQ)EBC3087619
 (EXLCZ)993400000000101689
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