Integrable Hamiltonian Systems: Geometry, Topology, Classification, 1st Edition (Hardback) book cover

Integrable Hamiltonian Systems

Geometry, Topology, Classification, 1st Edition

By A.V. Bolsinov, A.T. Fomenko

CRC Press

752 pages | 391 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9780415298056
pub: 2004-02-25
$185.00
x
eBook (VitalSource) : 9780429212406
pub: 2004-02-25
from $28.98


FREE Standard Shipping!

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, both of whom have contributed significantly to the field, develop the classification theory for integrable systems with two degrees of freedom. This theory allows one to distinguish such systems up to two natural equivalence relations: the equivalence of the associated foliation into Liouville tori and the usual orbital equaivalence. The authors show that in both cases, one can find complete sets of invariants that give the solution of the classification problem.

The first part of the book systematically presents the general construction of these invariants, including many examples and applications. In the second part, the authors apply the general methods of the classification theory to the classical integrable problems in rigid body dynamics and describe their topological portraits, bifurcations of Liouville tori, and local and global topological invariants. They show how the classification theory helps find hidden isomorphisms between integrable systems and present as an example their proof that two famous systems--the Euler case in rigid body dynamics and the Jacobi problem of geodesics on the ellipsoid--are orbitally equivalent.

Integrable Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.

Reviews

"This book provides an introduction to the problem of classification of integrable Hamiltonian systems. It presents, in a systematic way, a great deal of material previously available only in journals."

- Zentralblatt MATH, 1056

Table of Contents

BASIC NOTIONS

Linear Symplectic Geometry

Symplectic and Poisson Manifolds

The Darboux Theorem

Liouville Integrable Hamiltonian Systems. The Liouville Theorem

Non-Resonant and Resonant Systems

Rotation Number

The Momentum Mapping of an Integrable System and Its Bifurcation Diagram

Non-Degenerate Critical Points of the Momentum Mapping

Main Types of Equivalence of Dynamical Systems

THE TOPOLOGY OF FOLIATIONS ON TWO-DIMENSIONAL SURFACES

Generated by Morse Functions

Simple Morse Functions

Reeb Graph of a Morse Function

Notion of an Atom

Simple Atoms

Simple Molecules

Complicated Atoms

Classification of Atoms

Symmetry Groups of Oriented Atoms and the Universal Covering Tree

Notion of a Molecule

Approximation of Complicated Molecules by Simple Ones

Classification of Morse-Smale Flows on Two-Dimensional Surfaces by Means of Atoms and Molecules

ROUGH LIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM

Classification of Non-degenerate Critical Submanifolds on Isoenergy 3-Surfaces

The Topological Structure of a Neighborhood of a Singular Leaf

Topologically Stable Hamiltonian Systems

Example of a Topologically Unstable Integrable System

2-Atoms and 3-Atoms

Classification of 3-Atoms

3-Atoms as Bifurcations of Liouville Tori

The Molecule of an Integrable System

Complexity of Integrable Systems

LIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM

Admissible Coordinate Systems on the Boundary of a 3-Atom

Gluing Matrices and Superfluous Frames

Invariants (Numerical Marks) r, e, and n

The Marked Molecule is a Complete Invariant of Liouville Equivalence

The Influence of the Orientation

Realization Theorem

Simple Examples of Molecules

Hamiltonian Systems with Critical Klein Bottles

Topological Obstructions to Integrability of Hamiltonian Systems with Two Degrees of Freedom

ORBITAL CLASSIFICATION OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM

Rotation Function and Rotation Vector

Reduction of the Three-Dimensional Orbital Classification to the Two-Dimensional Classification up to Conjugacy

General Concept of Constructing Orbital Invariants of Integrable Hamiltonian Systems

CLASSIFICATION OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACES UP TO TOPOLOGICAL CONJUGACY

Invariants of a Hamiltonian System on a 2-Atom

Classification of Hamiltonian Flows with One Degree of Freedom up to Topological Conjugacy

Classification of Hamiltonian Flows on 2-Atoms with Involution up to Topological Conjugacy

The Pasting-Cutting Operation

Description of the Sets of Admissible delta-Invariants and Z-Invariants

SMOOTH CONJUGACY OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACES

Constructing Smooth Invariants on 2-Atoms

Theorem of Classification of Hamiltonian Flows on Atoms up to Smooth Conjugacy

ORBITAL CLASSIFICATION OF INTEGRABLE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. THE SECOND STEP

Superfluous t-Frame of a Molecule (Topological Case). The Main Lemma on t-Frames

The Group of Transformations of Transversal Sections. Pasting-Cutting Operation

The Action of GP on the Set of Superfluous t-Frames

Three General Principles for Constructing Invariants

Admissible Superfluous t-Frames and a Realization Theorem

Construction of Orbital Invariants in the Topological Case. A t-Molecule

Theorem on the Topological Orbital Classification of Integrable Systems with Two Degrees of Freedom

A Particular Case: Simple Integrable Systems

Smooth Orbital Classification

LIOUVILLE CLASSIFICATION OF INTEGRABLE SYSTEMS WITH NEIGHBORHOODS OF SINGULAR POINTS

l-Type of a Four-Dimensional Singularity

The Loop Molecule of a Four-Dimensional Singularity

Center-Center Case

Center-Saddle Case

Saddle-Saddle Case

Almost Direct Product Representation of a Four-Dimensional Singularity

Proof of the Classification Theorems

Focus-Focus Case

Almost Direct Product Representation for Multidimensional Non-degenerate Singularities of Liouville Foliations

METHODS OF CALCULATION OF TOPOLOGICAL INVARIANTS OF INTEGRABLE HAMILTONIAN SYSTEMS

General Scheme for Topological Analysis of the Liouville Foliation

Methods for Computing Marks

The Loop Molecule Method

List of Typical Loop Molecules

The Structure of the Liouville Foliation for Typical Degenerate Singularities

Typical Loop Molecules Corresponding to Degenerate One-Dimensional Orbits

Computation of r- and e-Marks by Means of Rotation Functions

Computation of the n-Mark by Means of Rotation Functions

Relationship Between the Marks of the Molecule and the Topology of Q3

INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES 409

Statement of the Problem

Topological Obstructions to Integrability of Geodesic Flows on Two-Dimensional Surfaces

Two Examples of Integrable Geodesic Flows

Riemannian Metrics Whose Geodesic Flows are Integrable by Means of Linear or Quadratic Integrals. Local Theory

Linearly and Quadratically Integrable Geodesic Flows on Closed Surfaces

LIOUVILLE CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES

The Torus

The Klein Bottle

The Sphere

The Projective Plane

ORBITAL CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES

Case of the Torus

Case of the Sphere

Examples of Integrable Geodesic Flows on the Sphere

Non-triviality of Orbital Equivalence Classes and Metrics with Closed Geodesics

THE TOPOLOGY OF LIOUVILLE FOLIATIONS IN CLASSICAL INTEGRABLE CASES IN RIGID BODY DYNAMICS

Integrable Cases in Rigid Body Dynamics

Topological Type of Isoenergy 3-Surfaces

Liouville Classification of Systems in the Euler Case

Liouville Classification of Systems in the Lagrange Case

Liouville Classification of Systems in the Kovalevskaya Case

Liouville Classification of Systems in the Goryachev-Chaplygin-Sretenskii Case

Liouville Classification of Systems in the Zhukovskii Case

Rough Liouville Classification of Systems in the Clebsch Case

Rough Liouville Classification of Systems in the Steklov Case

Rough Liouville Classification of Integrable Four-Dimensional Rigid Body Systems

The Complete List of Molecules Appearing in Integrable Cases of Rigid Body Dynamics

MAUPERTUIS PRINCIPLE AND GEODESIC EQUIVALENCE

General Maupertuis Principle

Maupertuis Principle in Rigid Body Dynamics

Classical Cases of Integrability in Rigid Body Dynamics and Related Integrable Geodesic Flows on the Sphere

Conjecture on Geodesic Flows with Integrals of High Degree

Dini Theorem and the Geodesic Equivalence of Riemannian

Metrics

Generalized Dini-Maupertuis Principle

Orbital Equivalence of the Neumann Problem and the Jacobi Problem

Explicit Forms of Some Remarkable Hamiltonians and Their Integrals in Separating Variables

EULER CASE IN RIGID BODY DYNAMICS AND JACOBI PROBLEM ABOUT GEODESICS ON THE ELLIPSOID. ORBITAL ISOMORPHISM

Introduction

Jacobi Problem and Euler Case

Liouville Foliations

Rotation Functions

The Main Theorem

Smooth Invariants

Topological Non-Conjugacy of the Jacobi Problem and the Euler Case

REFERENCES

SUBJECT INDEX

Subject Categories

BISAC Subject Codes/Headings:
MAT007000
MATHEMATICS / Differential Equations
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis