Integrability and Nonintegrability in Geometry and Mechanics
Springer (Verlag)
978-94-010-7880-1 (ISBN)
1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- §1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- §2. Symplectic Manifolds.- §3. Hamiltonian Properties of the Equations of Motion of a Three-Dimensional Rigid Body.- §4. Some Information on Lie Groups and Lie Algebras Necessary for Hamiltonian Geometry.- 2. The Theory of Surgery on Completely Integrable Hamiltonian Systems of Differential Equations.- §1. Classification of Constant-Energy Surfaces of Integrable Systems. Estimation of the Amount of Stable Periodic Solutions on a Constant-Energy Surface. Obstacles in the Way of Smooth Integrability of Hamiltonian Systems.- §2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- §3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- §1. Noncommutative Integration Method.- §2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- §3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- §4. Liouville Integrability on Complex Symplectic Manifolds.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- §1. Lie Algebras and Mechanics.- §2. Integrable Multidimensional Analogues of Mechanical Systems Whose Quadratic Hamiltonians are Contained in the Discovered Maximal Linear Commutative Algebras of Polynomials on Orbits of Lie Algebras.- §3. Euler Equations on the Lie Algebra so(4).- §4. Duplication of Integrable Analogues of the Euler Equationsby Means of Associative Algebra with Poincaré Duality.- §5. The Orbit Method in Hamiltonian Mechanics and Spin Dynamics of Superfluid Helium-3.- 5. Nonintegrability of Certain Classical Hamiltonian Systems.- §1. The Proof of Nonintegrability by the Poincaré Method.- §2. Topological Obstacles for Complete Integrability.- §3. Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds.- §4. Integrability and Nonintegrability of Geodesic Flows on Two-Dimensional Surfaces, Spheres, and Tori.- 6. A New Topological Invariant of Hamiltonian Systems of Liouville-Integrable Differential Equations. An Invariant Portrait of Integrable Equations and Hamiltonians.- §1. Construction of the Topological Invariant.- §2. Calculation of Topological Invariants of Certain Classical Mechanical Systems.- §3. Morse-Type Theory for Hamiltonian Systems Integrated by Means of Non-Bott Integrals.- References.
Reihe/Serie | Mathematics and its Applications ; 31 | Mathematics and its Applications ; 31 |
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Zusatzinfo | XV, 343 p. |
Verlagsort | Dordrecht |
Sprache | englisch |
Maße | 155 x 235 mm |
Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
Naturwissenschaften ► Physik / Astronomie | |
ISBN-10 | 94-010-7880-7 / 9401078807 |
ISBN-13 | 978-94-010-7880-1 / 9789401078801 |
Zustand | Neuware |
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