Author | Fomenko, A. T. author |
---|---|
Title | Integrability and Nonintegrability in Geometry and Mechanics [electronic resource] / by A. T. Fomenko |
Imprint | Dordrecht : Springer Netherlands, 1988 |
Connect to | http://dx.doi.org/10.1007/978-94-009-3069-8 |
Descript | XV, 343 p. online resource |
1. Some Equations of Classical Mechanics and Their Hamiltonian Properties -- ยง1. Classical Equations of Motion of a Three-Dimensional Rigid Body -- ยง2. Symplectic Manifolds -- ยง3. Hamiltonian Properties of the Equations of Motion of a Three-Dimensional Rigid Body -- ยง4. 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 -- ยง1. 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 -- ยง2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams -- ยง3. 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 -- ยง1. Noncommutative Integration Method -- ยง2. The General Properties of Invariant Submanifolds of Hamiltonian Systems -- ยง3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense -- ยง4. Liouville Integrability on Complex Symplectic Manifolds -- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications -- ยง1. Lie Algebras and Mechanics -- ยง2. 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 -- ยง3. Euler Equations on the Lie Algebra so(4) -- ยง4. Duplication of Integrable Analogues of the Euler Equations by Means of Associative Algebra with Poincarรฉ Duality -- ยง5. The Orbit Method in Hamiltonian Mechanics and Spin Dynamics of Superfluid Helium-3 -- 5. Nonintegrability of Certain Classical Hamiltonian Systems -- ยง1. The Proof of Nonintegrability by the Poincarรฉ Method -- ยง2. Topological Obstacles for Complete Integrability -- ยง3. Topological Obstacles for Analytic Integrability of Geodesic Flows on Non-Simply-Connected Manifolds -- ยง4. 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 -- ยง1. Construction of the Topological Invariant -- ยง2. Calculation of Topological Invariants of Certain Classical Mechanical Systems -- ยง3. Morse-Type Theory for Hamiltonian Systems Integrated by Means of Non-Bott Integrals -- References