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141009  eng 
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a 9783034807234

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1 

a Morales Ruiz, Juan J.

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a Differential Galois Theory and NonIntegrability of Hamiltonian Systems
h Elektronische Ressource
c by Juan J. Morales Ruiz

250 


a 1st ed. 1999

260 


a Basel
b Birkhäuser
c 1999, 1999

300 


a XIV, 167 p. 5 illus
b online resource

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0 

a 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

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a Algebraic fields

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a Field Theory and Polynomials

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a Manifolds (Mathematics)

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a Differential Equations

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a Polynomials

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a Global analysis (Mathematics)

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a Global Analysis and Analysis on Manifolds

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a Differential equations

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0 
7 
a eng
2 ISO 6392

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b SBA
a Springer Book Archives 2004

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a Modern Birkhäuser Classics

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a 10.1007/9783034887182

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u https://doi.org/10.1007/9783034887182?nosfx=y
x Verlag
3 Volltext

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a 515.35

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a    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 wideapplied 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)

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a 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.

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a 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.
