Hamiltonian Systems And Their Integrability
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Author | : Mich'le Audin |
Publisher | : American Mathematical Soc. |
Total Pages | : 172 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 9780821844137 |
Download Hamiltonian Systems and Their Integrability Book in PDF, Epub and Kindle
"This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations. This book would be suitable for a graduate course in Hamiltonian systems."--BOOK JACKET.
Author | : Juan J. Morales Ruiz |
Publisher | : Birkhäuser |
Total Pages | : 177 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034887183 |
Download Differential Galois Theory and Non-Integrability of Hamiltonian Systems Book in PDF, Epub and Kindle
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)
Author | : Vladimir Gerdjikov |
Publisher | : Springer Science & Business Media |
Total Pages | : 645 |
Release | : 2008-06-02 |
Genre | : Science |
ISBN | : 3540770534 |
Download Integrable Hamiltonian Hierarchies Book in PDF, Epub and Kindle
This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their hierarchies.
Author | : A.V. Bolsinov |
Publisher | : CRC Press |
Total Pages | : 752 |
Release | : 2004-02-25 |
Genre | : Mathematics |
ISBN | : 0203643429 |
Download Integrable Hamiltonian Systems Book in PDF, Epub and Kindle
Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors,
Author | : Michèle Audin |
Publisher | : Birkhäuser |
Total Pages | : 225 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3034880715 |
Download Symplectic Geometry of Integrable Hamiltonian Systems Book in PDF, Epub and Kindle
Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. This book serves as an introduction to symplectic and contact geometry for graduate students, exploring the underlying geometry of integrable Hamiltonian systems. Includes exercises designed to complement the expositiont, and up-to-date references.
Author | : Alain Goriely |
Publisher | : World Scientific |
Total Pages | : 438 |
Release | : 2001 |
Genre | : Science |
ISBN | : 9789812811943 |
Download Integrability and Nonintegrability of Dynamical Systems Book in PDF, Epub and Kindle
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.
Author | : Velimir Jurdjevic |
Publisher | : American Mathematical Soc. |
Total Pages | : 150 |
Release | : 2005 |
Genre | : Mathematics |
ISBN | : 0821837648 |
Download Integrable Hamiltonian Systems on Complex Lie Groups Book in PDF, Epub and Kindle
Studies the elastic problems on simply connected manifolds $M_n$ whose orthonormal frame bundle is a Lie group $G$. This title synthesizes ideas from optimal control theory, adapted to variational problems on the principal bundles of Riemannian spaces, and the symplectic geometry of the Lie algebra $\mathfrak{g}, $ of $G$
Author | : A.T. Fomenko |
Publisher | : Springer Science & Business Media |
Total Pages | : 358 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9400930690 |
Download Integrability and Nonintegrability in Geometry and Mechanics Book in PDF, Epub and Kindle
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . • 1111 Oulik'. n. . Chi" •. • ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Author | : Lawrence Markus |
Publisher | : American Mathematical Soc. |
Total Pages | : 58 |
Release | : 1974 |
Genre | : Differential equations |
ISBN | : 0821818449 |
Download Generic Hamiltonian Dynamical Systems are Neither Integrable nor Ergodic Book in PDF, Epub and Kindle
This memoir gives an introduction to Hamiltonian dynamical systems on symplectic manifolds, including definitions of Hamiltonian vector fields, Poisson brackets, integrals of motion, complete integrability, and ergodicity. A particularly complete treatment of action-angle coordinates is given. Historical background into the question of ergodicity and integrability in Hamiltonian systems is also given.
Author | : Richard H. Cushman |
Publisher | : Birkhäuser |
Total Pages | : 449 |
Release | : 2012-12-06 |
Genre | : Science |
ISBN | : 3034888910 |
Download Global Aspects of Classical Integrable Systems Book in PDF, Epub and Kindle
This book gives a complete global geometric description of the motion of the two di mensional hannonic oscillator, the Kepler problem, the Euler top, the spherical pendulum and the Lagrange top. These classical integrable Hamiltonian systems one sees treated in almost every physics book on classical mechanics. So why is this book necessary? The answer is that the standard treatments are not complete. For instance in physics books one cannot see the monodromy in the spherical pendulum from its explicit solution in terms of elliptic functions nor can one read off from the explicit solution the fact that a tennis racket makes a near half twist when it is tossed so as to spin nearly about its intermediate axis. Modem mathematics books on mechanics do not use the symplectic geometric tools they develop to treat the qualitative features of these problems either. One reason for this is that their basic tool for removing symmetries of Hamiltonian systems, called regular reduction, is not general enough to handle removal of the symmetries which occur in the spherical pendulum or in the Lagrange top. For these symmetries one needs singular reduction. Another reason is that the obstructions to making local action angle coordinates global such as monodromy were not known when these works were written.