Integrability and Nonintegrability of Dynamical Systems

2001 Science

Author : Alain Goriely
Release : 2001
Genre : Science
Kind : eBook
Book Rating : 943/5 ( reviews)

Book Synopsis Integrability and Nonintegrability of Dynamical Systems by : Alain Goriely

Download or read book Integrability and Nonintegrability of Dynamical Systems written by Alain Goriely. This book was released on 2001. Available in PDF, EPUB and Kindle. Book excerpt: This invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems). As generic systems of differential equations cannot be exactly solved, the book reviews the different notions of nonintegrability and shows how to prove the nonexistence of exact solutions and/or a constant of motion. Finally, nonintegrability theory is linked to dynamical systems theory by showing how the property of complete integrability, partial integrability or nonintegrability can be related to regular and irregular dynamics in phase space. Contents: Integrability: An Algebraic Approach; Integrability: An Analytic Approach; Polynomial and Quasi-Polynomial Vector Fields; Nonintegrability; Hamiltonian Systems; Nearly Integrable Dynamical Systems; Open Problems. Readership: Mathematical and theoretical physicists and astronomers and engineers interested in dynamical systems.

Differential Galois Theory and Non-Integrability of Hamiltonian Systems

2012-12-06 Mathematics

Author : Juan J. Morales Ruiz
Release : 2012-12-06
Genre : Mathematics
Kind : eBook
Book Rating : 183/5 ( reviews)

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

Download or read book Differential Galois Theory and Non-Integrability of Hamiltonian Systems written by Juan J. Morales Ruiz. This book was released on 2012-12-06. Available in PDF, EPUB and Kindle. Book excerpt: 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 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, 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. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and 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) 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)

Integrability of Dynamical Systems: Algebra and Analysis

2017-03-30 Mathematics

Author : Xiang Zhang
Release : 2017-03-30
Genre : Mathematics
Kind : eBook
Book Rating : 268/5 ( reviews)

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Book Synopsis Integrability of Dynamical Systems: Algebra and Analysis by : Xiang Zhang

Download or read book Integrability of Dynamical Systems: Algebra and Analysis written by Xiang Zhang. This book was released on 2017-03-30. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

Symmetries and Singularity Structures

1990 Mathematics

Author : Muthuswamy Lakshmanan
Release : 1990
Genre : Mathematics
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis Symmetries and Singularity Structures by : Muthuswamy Lakshmanan

Download or read book Symmetries and Singularity Structures written by Muthuswamy Lakshmanan. This book was released on 1990. Available in PDF, EPUB and Kindle. Book excerpt: Symmetries and singularity structures play important roles in the study of nonlinear dynamical systems. It was Sophus Lie who originally stressed the importance of symmetries and invariance in the study of nonlinear differential equations. How ever, the full potentialities of symmetries had been realized only after the advent of solitons in 1965. It is now a well-accepted fact that associated with the infinite number of integrals of motion of a given soliton system, an infinite number of gep. eralized Lie BAcklund symmetries exist. The associated bi-Hamiltonian struc ture, Kac-Moody, Vrrasoro algebras, and so on, have been increasingly attracting the attention of scientists working in this area. Similarly, in recent times the role of symmetries in analyzing both the classical and quantum integrable and nonintegrable finite dimensional systems has been remarkable. On the other hand, the works of Fuchs, Kovalevskaya, Painleve and coworkers on the singularity structures associated with the solutions of nonlinear differen tial equations have helped to classify first and second order nonlinear ordinary differential equations. The recent work of Ablowitz, Ramani and Segur, con jecturing a connection between soliton systems and Painleve equations that are free from movable critical points, has motivated considerably the search for the connection between integrable dynamical systems with finite degrees of freedom and the Painleve property. Weiss, Tabor and Carnevale have extended these ideas to partial differential equations."

Dynamical Systems VII

2013-12-14 Mathematics

Author : V.I. Arnol'd
Release : 2013-12-14
Genre : Mathematics
Kind : eBook
Book Rating : 96X/5 ( reviews)

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Book Synopsis Dynamical Systems VII by : V.I. Arnol'd

Download or read book Dynamical Systems VII written by V.I. Arnol'd. This book was released on 2013-12-14. Available in PDF, EPUB and Kindle. Book excerpt: A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

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Integrability and Nonintegrability of Dynamical Systems

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