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Author: Michèle Audin
Publisher: Birkhäuser
ISBN: 3034880715
Category : Mathematics
Languages : en
Pages : 225
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Book Description
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: Michèle Audin
Publisher: Birkhäuser
ISBN: 3034880715
Category : Mathematics
Languages : en
Pages : 225
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Book Description
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: Tudor Ratiu
Publisher: Springer Science & Business Media
ISBN: 1461397251
Category : Mathematics
Languages : en
Pages : 526
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Book Description
The papers in this volume are an outgrowth of the lectures and informal discussions that took place during the workshop on "The Geometry of Hamiltonian Systems" which was held at MSRl from June 5 to 16, 1989. It was, in some sense, the last major event of the year-long program on Symplectic Geometry and Mechanics. The emphasis of all the talks was on Hamiltonian dynamics and its relationship to several aspects of symplectic geometry and topology, mechanics, and dynamical systems in general. The organizers of the conference were R. Devaney (co-chairman), H. Flaschka (co-chairman), K. Meyer, and T. Ratiu. The entire meeting was built around two mini-courses of five lectures each and a series of two expository lectures. The first of the mini-courses was given by A. T. Fomenko, who presented the work of his group at Moscow University on the classification of integrable systems. The second mini course was given by J. Marsden of UC Berkeley, who spoke about several applications of symplectic and Poisson reduction to problems in stability, normal forms, and symmetric Hamiltonian bifurcation theory. Finally, the two expository talks were given by A. Fathi of the University of Florida who concentrated on the links between symplectic geometry, dynamical systems, and Teichmiiller theory.
Author: Alexey Bolsinov
Publisher: Birkhäuser
ISBN: 3319335030
Category : Mathematics
Languages : en
Pages : 148
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Book Description
Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields. Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mirror symmetry). As such, the book will appeal to experts with a wide range of backgrounds.
Author: A.V. Bolsinov
Publisher: CRC Press
ISBN: 0203643429
Category : Mathematics
Languages : en
Pages : 752
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Book Description
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: A.T. Fomenko
Publisher: CRC Press
ISBN: 9782881249013
Category : Mathematics
Languages : en
Pages : 488
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Book Description
Author: Velimir Jurdjevic
Publisher: Cambridge University Press
ISBN: 1316586332
Category : Mathematics
Languages : en
Pages : 437
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Book Description
The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.
Author: Kang Feng
Publisher: Springer Science & Business Media
ISBN: 3642017770
Category : Mathematics
Languages : en
Pages : 690
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Book Description
"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.
Author: Vladimir Gerdjikov
Publisher: Springer Science & Business Media
ISBN: 3540770534
Category : Science
Languages : en
Pages : 645
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Book Description
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: Pavle Saksida
Publisher:
ISBN:
Category : Hamiltonian systems
Languages : en
Pages : 300
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Book Description
The topic of this thesis are finite dimensional Hamiltonian integrable systems and certain aspects of the symplectic geometry of their underlying phase spaces. The main result is presented in Chapter 3. The complete integrability of a class of hamiltonian systems $T^{\ast}M,\;\omega_{can},\; H)$ is proved, where $M$ is an arbitrary compact or non-compact Riemannian symmetric space. This class contains some classical examples of integrable systems e.g. the C. Neumann's system, and the spherical pendulum. The new examples we consider are motions on projective spaces, which in turn yield integrable motions of a particle on {\bf CP}$^n$ and {\bd HP}$^n$ in a quadratic potential field with the additional presence of the magnetic and the Yang-Mills fields respectively. The connection of the systems $T^{\ast}M,\;\omega_{can},\;H)$ with The Nahm's equations is investigated. It is also indicated, how these systems fit into the context of the Hitchin's integrable systems on $T^{\ast}{\cal M}_{par}$, where ${\cal M}_{par}$ is the moduli space of stable parabolic structures on $G^{\CC}$-principal bundle over a complex curve $C$. Hitchin's systems on $T^{\ast}{\cal M}_{par}$ are studied in tha Chapter 2. In a different way, these systens were alresdy constructed by E. Markman. Our approach allows us to show how the weights of ${\cal M}_{par}$, perturb the symplectid structureof $T^{\ast}{\cal M }_{par}$ by adding a magnetic term $\gamma_{\lambda (D)}$ to the canonical structure $\omega_{can}$ on $T^{\ast}{\cal M}_{par}$. The terms are parametrised by $D\times Lie(K)$ where $D$ is the divisor of marked poins and $Lie(K)$ Cartan sub-algebra of $Lie(G^{\CC})$. The Hitchin's systems can be thought of as describing a motion of a particle on ${\cal M}_{par}$ influenced by certain forces, where the term $\gamma_{\lambda (D)}$ gives rise to a force of magnetic type. The variations of the symplectic structure on $T^{\ast}{\cal M}_{par}$, stem from an aspect of the symplectic geometry of a complex coadjoint orbit ${\cal O}^{\CC}$, which is the main topic of the Chapter 1. The orbit ${\cal O}^{\CC}$ has the natural Konstant-Kirillov symplectic structure, and the canonical cotangent structure since ${\cal O}^{\CC}=T^{\ast}{\cal O}$ for a compact orbit ${\cal O}$. The two structures differ by a magnetic term coming from a suitable mechanical connection. The construction is generalised to the case where ${\cal O}^{\CC}$ is replaced by ${\cal O}^{\CC}$-fibre bundle.
Author: Mich'le Audin
Publisher: American Mathematical Soc.
ISBN: 9780821844137
Category : Mathematics
Languages : en
Pages : 172
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Book Description
"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.
