Symmetries, Topology and Resonances in Hamiltonian Mechanics

I Hamiltonian Mechanics.- 1 The Hamilton Equations.- 2 Euler-Poincaré 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 Poincaré Method.- 2 Applications of the Poincaré 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 Kirchhoff’s 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 Siegel’s 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 Poincaré 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.