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Hamiltonian Systems: Geometry, Topology, Classification offers a unique opportunity to explore important, previously unpublished results and acquire generally applicable techniques and tools that enable you to work with a broad class of integrable systems.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