Integrable Hamiltonian systems : geometry, topology, classification /: geometry, topology, classification. (©2004)

• Record Type: Book
• Uniform Title: Integriruemye gamiltonovy sistemy.
• Further Information: Note: A.V. Bolsinov and A.T. Fomenko.
• Other Names: Bolsinov, A. V (Alekse Viktorovich); Fomenko, A. T
• 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
• Publisher Details: Boca Raton, Fla : Chapman & Hall/CRC
• Publication Date: 2004
• Copyright Date: 2004
• Extent: 1 online resource (xv, 730 pages), illustrations
• Subjects: 515/.39; Geodesic flows; Geodesics (Mathematics); Hamiltonian systems; MATHEMATICS -- Differential Equations -- General; Geodesic flows; Geodesics (Mathematics); Hamiltonian systems; Electronic books
• Languages: English
• ISBNs: 0415298059; 9780415298056; 0203643429; 9780203643426
• Notes: Note: Includes bibliographical references (pages 705-723) and index.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.