Symmetries, Topology and Resonances in Hamiltonian Mechanics

John Hornstein has written about the author's theorem on nonintegrability of geodesic flows on closed surfaces of genus greater than one: "Here is an example of how differential geometry, differential and algebraic topology, and Newton's laws make music together" (Amer. Math. Monthly, November 1989). Kozlov's book is a systematic introduction to the problem of exact integration of equations of dynamics. The key to the solution is to find nontrivial symmetries of Hamiltonian systems. After Poincaré's work it became clear that topological considerations and the analysis of resonance phenomena play a crucial role in the problem on the existence of symmetry fields and nontrivial conservation laws.

Table of Contents:
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.