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Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects)
L. M. Lerman, Research Institute for Applied Mathematics and Cybernetics, Nizhni Novgorod, Russia, and Ya. L. Umanskiy, Total System Services, Inc., Atlanta, GA

Translations of Mathematical Monographs
1998; 177 pp; hardcover
Volume: 176
ISBN-10: 0-8218-0375-1
ISBN-13: 978-0-8218-0375-2
List Price: US$99
Member Price: US$79.20
Order Code: MMONO/176
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The main topic of this book is the isoenergetic structure of the Liouville foliation generated by an integrable system with two degrees of freedom and the topological structure of the corresponding Poisson action of the group \({\mathbb R}^2\). This is a first step towards understanding the global dynamics of Hamiltonian systems and applying perturbation methods. Emphasis is placed on the topology of this foliation rather than on analytic representation. In contrast to previously published works in this area, here the authors consistently use the dynamical properties of the action to achieve their results.


Graduate students and research mathematicians working in the dynamics of Hamiltonian systems; also useful for those studying the geometric structure of symplectic manifolds.


"The main goal of the book is to obtain isoenergetic equivalence of IHVFs in some special neighborhoods of a simple singular point. Therefore, in the following chapters, the authors consider each possible type of singular point separately: elliptic, saddle-center, saddle and focus-saddle. Various examples of each case are presented in the last chapter. The interest of the book is that it concentrates on topological aspects of the subject rather than using an analytic point of view. In contrast to most of the books published previously, dynamical properties of the Poisson action are consistently used in order to achieve the results. This book can be used by graduate students and researchers interested in studying dynamics of Hamiltonian systems. It can also be useful for people studying the geometric structure of symplectic manifolds."

-- Mathematical Reviews

Table of Contents

  • General results of the theory of Hamiltonian systems
  • Linear theory and classification of singular orbits
  • IHVF and Poisson actions of Morse type
  • Center-center type singular points of PA and elliptic singular points of IHVF
  • Saddle-center type singular points
  • Saddle type singular points
  • Saddle-focus type singular points
  • Realization
  • Normal forms of quadratic Hamilton functions and their centralizers in \(sp(4,{\mathbb R})\)
  • The gradient system on \(M\) compatible with the Hamiltonian
  • Bibliography
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