**Translations of Mathematical Monographs**

1998;
177 pp;
Hardcover

MSC: Primary 58; 70;

**Print ISBN: 978-0-8218-0375-2
Product Code: MMONO/176**

List Price: $105.00

AMS Member Price: $84.00

MAA Member Price: $94.50

**Electronic ISBN: 978-1-4704-4591-1
Product Code: MMONO/176.E**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

# Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects)

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*L. M. Lerman; Ya. L. Umanskiy*

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.

#### Readership

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

#### Reviews & Endorsements

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

# Table of Contents

## Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects)

- Cover Cover11
- Title page v6
- Contents vii8
- Introduction ix10
- General results of the theory of Hamiltonian systems 114
- Linear theory and classification of singular orbits 1730
- IHVF and Poisson actions of Morse type 4356
- Center-center type singular points of PA and elliptic singular points of IHVF 5164
- Saddle-center type singular points 5770
- Saddle type singular points 7184
- Saddle-focus type singular points 129142
- Realization 149162
- Normal forms of quadratic Hamilton functions and their centralizers in ๐ ๐(4,โ) 163176
- The gradient system on ๐ compatible with the Hamiltonian 167180
- Bibliography 175188
- Back Cover Back Cover1193