Author | Kozlov, Valerij V. author |
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Title | Symmetries, Topology and Resonances in Hamiltonian Mechanics [electronic resource] / by Valerij V. Kozlov |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1996 |
Connect to | http://dx.doi.org/10.1007/978-3-642-78393-7 |
Descript | XI, 378 p. online resource |
I Hamiltonian Mechanics -- 1 The Hamilton Equations -- 2 Euler-Poincarรฉ Equations on Lie Algebras -- 3 The Motion of a Rigid Body -- 4 Pendulum Oscillations -- 5 Some Problems of Celestial Mechanics -- 6 Systems of Interacting Particles -- 7 Non-holonomic Systems -- 8 Some Problems of Mathematical Physics -- 9 The Problem of Identification of Hamiltonian Systems -- II Integration of Hamiltonian Systems -- 1 Integrals. Classes of Integrals of Hamiltonian Systems -- 2 Invariant Relations -- 3 Symmetry Groups -- 4 Complete Integrability -- 5 Examples of Completely Integrable Systems -- 6 Isomorphisms of Some Integrable Hamiltonian Systems -- 7 Separation of Variables -- 8 The Heisenberg Representation -- 9 Algebraically Integrable Systems -- 10 Perturbation Theory -- 11 Normal Forms -- III Topological and Geometrical Obstructions to Complete Integrability -- 1 Topology of the Configuration Space of an Integrable System -- 2 Proof of Nonintegrability Theorems -- 3 Geometrical Obstructions to Integrability -- 4 Systems with Gyroscopic Forces -- 5 Generic Integrals -- 6 Topological Obstructions to the Existence of Linear Integrals -- 7 Topology of the Configuration Space of Reversible Systems with Nontrivial Symmetry Groups -- IV Nonintegrability of Hamiltonian Systems Close to Integrable Ones -- 1 The Poincarรฉ Method -- 2 Applications of the Poincarรฉ Method -- 3 Symmetry Groups -- 4 Reversible Systems With a Torus as the Configuration Space -- 5 A Criterion for Integrability in the Case When the Potential is a Trigonometric Polynomial -- 6 Some Generalizations -- 7 Systems of Interacting Particles -- 8 Birth of Isolated Periodic Solutions as an Obstacle to Integrability -- 9 Non-degenerate Invariant Tori -- 10 Birth of Hyperbolic Invariant Tori -- 11 Non-Autonomous Systems -- V Splitting of Asymptotic Surfaces -- 1 Asymptotic Surfaces and Splitting Conditions -- 2 Theorems on Nonintegrability -- 3 Some Applications -- 4 Conditions for Nonintegrability of Kirchhoffโs Equations -- 5 Bifurcation of Separatrices -- 6 Splitting of Separatrices and Birth of Isolated Periodic Solutions -- 7 Asymptotic Surfaces of Unstable Equilibria -- 8 Symbolic Dynamics -- VI Nonintegrability in the Vicinity of an Equilibrium Position -- 1 Siegelโs Method -- 2 Nonintegrability of Reversible Systems -- 3 Nonintegrability of Systems Depending on Parameters -- 4 Symmetry Fields in the Vicinity of an Equilibrium Position -- VII Branching of Solutions and Nonexistence of Single-Valued Integrals -- 1 The Poincarรฉ Small Parameter Method -- 2 Branching of Solutions and Polynomial Integrals of Reversible Systems on a Torus -- 3 Integrals and Symmetry Groups of Quasi-Homogeneous Systems of Differential Equations -- 4 Kovalevskaya Numbers for Generalized Toda Lattices -- 5 Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals -- VIII Polynomial Integrals of Hamiltonian Systems -- 1 The Birkhoff Method -- 2 Influence of Gyroscopic Forces on the Existence of Polynomial Integrals -- 3 Polynomial Integrals of Systems with One and a Half Degrees of Freedom -- 4 Polynomial Integrals of Hamiltonian Systems with Exponential Interaction -- 5 Perturbations of Hamiltonian Systems with Non-Compact Invariant Surfaces -- References