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Author:   A.V. Bolsinov ,  A.T. Fomenko
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 2000
Dimensions:   Width: 17.80cm , Height: 1.80cm , Length: 25.40cm
Weight:   0.647kg
ISBN:  

9781461369332

ISBN 10:   1461369339
Pages:   322
Publication Date:   02 November 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

1. Basic Notions.- 1.1. Linear Symplectic Geometry.- 1.2. Symplectic and Poisson Manifolds.- 1.3. Local Properties of Symplectic Manifolds.- 1.4. Liouville Integrable Hamiltonian Systems. Liouville Theorem.- 1.5. Nonresonant and Resonant Systems.- 1.6. Rotation Number.- 1.7. Momentum Mapping of an Integrable Hamiltonian System and Its Bifurcation Diagram.- 1.8. Nondegenerate Singularities of the Momentum Mapping and Bott Functions.- 1.9. Bott Integrals from the Point of View of the Four-Dimensional Symplectic Manifold.- 1.10. Main Types of Equivalence of Dynamical Systems.- 2. Topology of Foliations Generated by Morse Functions on Two-Dimensional Surfaces.- 2.1. Simple Morse Functions.- 2.2. Reeb Graph of a Morse Function.- 2.3. Concept of an Atom.- 2.4. Simple Molecules.- 2.5. Complicated Atoms.- 2.6. Classification of Atoms.- 2.7. Notion of a Molecule.- 2.8. Approximation of Complicated Molecules by Simple Ones.- 3. Rough Liouville Equivalence of Integrable Systems with Two Degrees of Freedom.- 3.1. Classification of Nondegenerate Critical Submanifolds on Isoenergy 3-Surfaces.- 3.2. The Topological Structure of a Neighborhood of a Singular Leaf.- 3.3. Topologically Stable Hamiltonian Systems.- 3.4. 2-Atoms and 3-Atoms.- 3.5. Classification of 3-Atoms.- 3.6. 3-Atoms as Bifurcations of Liouville Tori.- 3.7. The Molecule of an Integrable System.- 4. Liouville Equivalence of Iintegrable Systems with Two Degrees of Freedom.- 4.1. Admissible Coordinate Systems on the Boundary of a 3-Atom.- 4.2. Gluing Matrices and Superfluous Frames.- 4.3. Invariants (Numerical Marks) r, ?, and n.- 4.4. The Marked Molecule.- 4.5. Influence of the Orientation.- 4.6. Realization Theorem.- 4.7. Simple Examples of Molecules.- 4.8. Hamiltonian Systems with Critical Klein Bottles.- 5. TrajectoryClassification of Integrable Systems with Two Degrees of Freedom.- 5.1. Rotation Function and Rotation Vector.- 5.2. Reduction of the Three-Dimensional.- 5.3. General Concept of Constructing Trajectory Invariants of Integrable Hamiltonian Systems.- 6. Integrable Geodesic Flows on Two-Dimensional Surfaces.- 6.1. Statement of the Problem.- 6.2. Topological Obstructions to Integrability of Geodesic Flows on Two-Dimensional Surfaces.- 6.3. Two Examples of Integrable Geodesic Flows.- 6.4. Riemannian Metrics Whose Geodesic Flows are Integrable by Means of Linear or Quadratic Integrals. Local Theory.- 6.5. Linearly and Quadratically Integrable Geodesic Flows on Closed Surfaces.- 7. Liouville Classification of Integrable Geodesic Flows on Two-Dimensional Surface.- 7.1. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Torus.- 7.2. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Klein Bottle.- 7.3. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Two-Dimensional Sphere.- 7.4. Liouville Classification of Linearly and Quadratically Integrable Geodesic Flows on the Projective Plane.- 8. Trajectory Classification of Integrable Geodesic Flows on Two-Dimensional Surfaces.- 8.1. Case of the Torus.- 8.2. Case of the Sphere.- 8.3. Examples of Integrable Geodesic Flows on the Sphere.- 8.4. Non-Triviality of Trajectory Equivalence Classes and Metrics with Closed Geodesics.- 9. Maupertuis Principle and Geodesic Equivalence.- 9.1. General Maupertuis Principle.- 9.2. Maupertuis Principle in Rigid Body Dynamics.- 9.3. Maupertuis Principle and an Explicit Form of the Metric on the Sphere, Generated by a Quadratic Hamiltonian on the Lie Algebra e(3).- 9.4. Classical Cases of Integrability in Rigid Body Dynamics and Related Integrable Geodesic Flows on the Sphere.- 9.5. Conjecture on Geodesic Flows with Integrals of High Degree.- 9.6. Dini Theorem and the Geodesic Equivalence of Riemannian Metrics.- 9.7. Generalized Dini-Maupertuis Principle.- 9.8. Trajectory Equivalence of the Neumann Problem and Jacobi Problem.- 9.9. Explicit Forms of Some Remarkable Hamiltonians and Their Integrals in Separating Variables.- 10. Euler Case in Rigid Body Dynamics and Jacobi Problem About Geodesics on the Ellipsoid. Trajectory Isomorphism.- 10.1. Introduction.- 10.2. Jacobi Problem and Euler Case.- 10.3. Liouville Foliations.- 10.4. Rotation Functions.- 10.5. The Main Theorem.- 10.6. Smooth Invariants.- 10.7. Topological Non-Conjugacy of the Jacobi Problem and the Euler Case.

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