Integrable Hamiltonian Systems: Geometry, Topology, Classification

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Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.

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