Integrability and Nonintegrability in Geometry and Mechanics

1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- §1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- 1.1. The Euler—Poisson Equations Describing the Motion of a Heavy Rigid Body around a Fixed Point.- 1.2. Integrable Euler, Lagrange, and Kovalevskaya Cases.- 1.3. General Equations of Motion of a Three-Dimensional Rigid Body.- §2. Symplectic Manifolds.- 2.1. Symplectic Structure in a Tangent Space to a Manifold.- 2.2. Symplectic Structure on a Manifold.- 2.3. Hamiltonian and Locally Hamiltonian Vector Fields and the Poisson Bracket.- 2.4. Integrals of Hamiltonian Fields.- 2.5. The Liouville Theorem.- §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.- 4.1. Adjoint and Coadjoint Representations, Semisimplicity, the System of Roots and Simple Roots, Orbits, and the Canonical Symplectic Structure.- 4.2. Model Example: SL(n, ?) and sl(n, ?).- 4.3. Real, Compact, and Normal Subalgebras.- 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.- 1.1. Formulation of the Results in Four Dimensions.- 1.2. A Short List of the Basic Data from the Classical Morse Theory.- 1.3. Topological Surgery on Liouville Tori of an Integrable Hamiltonian System upon Varying Values of a Second Integral.- 1.4. Separatrix Diagrams Cut out Nontrivial Cycles on Nonsingular Liouville Tori.- 1.5. The Topology of Hamiltonian-Level Surfaces of an Integrable System and of the Corresponding One-Dimensional Graphs.- 1.6. Proof of the Principal Classification Theorem 2.1.2.- 1.7. Proof of Claim 2.1.1.- 1.8.Proof of Theorem 2.1.1. Lower Estimates on the Number of Stable Periodic Solutions of a System.- 1.9. Proof of Corollary 2.1.5.- 1.10 Topological Obstacles for Smooth Integrability and Graphlike Manifolds. Not each Three-Dimensional Manifold Can be Realized as a Constant-Energy Manifold of an Integrable System.- 1.11. Proof of Claim 2.1.4.- §2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- 2.1. Bifurcation Diagram of the Momentum Mapping for an Integrable System. The Surgery of General Position.- 2.2. The Classification Theorem for Liouville Torus Surgery.- 2.3. Toric Handles. A Separatrix Diagram is Always Glued to a Nonsingular Liouville Torus Tn Along a Nontrivial (n — 1)-Dimensional Cycle Tn—1.- 2.4. Any Composition of Elementary Bifurcations (of Three Types) of Liouville Tori Is Realized for a Certain Integrable System on an Appropriate Symplectic Manifold.- 2.5. Classification of Nonintegrable Critical Submanifolds of Bott Integrals.- §3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3.1. A Fundamental Decomposition Q = mI +pII +qIII +sIV +rV and the Structure of Singular Fibres.- 3.2. Homological Properties of Constant-Energy Surfaces.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- §1. Noncommutative Integration Method.- 1.1. Maximal Linear Commutative Subalgebras in the Algebra of Functions on Symplectic Manifolds.- 1.2. A Hamiltonian System Is Integrable if Its Hamiltonian is Included in a Sufficiently Large Lie Algebra of Functions.- 1.3. Proof of the Theorem.- §2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- 2.1. Reduction of a System on One Isolated Level Surface.- 2.2. Further Generalizations of the Noncommutative Integration Method.- §3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- 3.1. The Formulation of the General Equivalence Hypothesis and its Validity for Compact Manifolds.- 3.2. The Properties of Momentum Mapping of a System Integrable in the Noncommutative Sense.- 3.3. Theorem on the Existence of Maximal Linear Commutative Algebras of Functions on Orbits in Semisimple and Reductive Lie Algebras.- 3.4. Proof of the Hypothesis for the Case of Compact Manifolds.- 3.5. Momentum Mapping of Systems Integrable in the Noncommutative Sense by Means of an Excessive Set of Integrals.- 3.6. Sufficient Conditions for Compactness of the Lie Algebra of Integrals of a Hamiltonian System.- §4. Liouville Integrability on Complex Symplectic Manifolds.- 4.1. Different Notions of Complex Integrability and Their Interrelation.- 4.2. Integrability on Complex Tori.- 4.3. Integrability on K3-Type Surfaces.- 4.4. Integrability on Beauville Manifolds.- 4.5.Symplectic Structures Integrated without Degeneracies.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- §1. Lie Algebras and Mechanics.- 1.1. Embeddings of Dynamic Systems into Lie Algebras.- 1.2. List of the Discovered Maximal Linear Commutative Algebras of Polynomials on the Orbits of Coadjoint Representations of Lie Groups.- §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.- 2.1. The Description of Integrable Quadratic Hamiltonians.- 2.2. Cases of Complete Integrability of Equations of Various Motions of a Rigid Body.- 2.3. Geometric Properties of Rigid-Body Invariant Metrics on Homogeneous Spaces.- §3. Euler Equations on the Lie Algebra so(4).- §4. Duplication of Integrable Analogues of the Euler Equations by Means of Associative Algebra with Poincaré Duality.- 4.1. Algorithm for Constructing Integrable Lie Algebras.- 4.2. Frobenius Algebras and Extensions of Lie Algebras.- 4.3. Maximal Linear Commutative Algebras of Functions on Contractions of Lie Algebras.- §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.- 1.1. Perturbation Theory and the Study of Systems Close to Integrable.- 1.2. Nonintegrability of the Equations of Motion of a Dynamically Nonsymmetric Rigid Body with a Fixed Point.- 1.3. Separatrix Splitting.- 1.4. Nonintegrability in the General Case of the Kirchhoff Equations of Motion of a Rigid Body in an Ideal Liquid.- §2. Topological Obstacles for Complete Integrability.- 2.1. Nonintegrability of the Equations of Motion of Natural Mechanical Systems with Two Degrees of Freedom on High-Genus Surfaces.- 2.2. Nonintegrability of Geodesic Flows on High-Genus Riemann Surfaces with Convex Boundary.- 2.3. Nonintegrability of the Problem of n Gravitating Centres for n > 2.- 2.4. Nonintegrability of Several Gyroscopic Systems.- §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.- 4.1. The Holomorphic 1-Form of the Integral of a Geodesic Flow Polynomial in Momenta and the Theorem on Nonintegrability of Geodesic Flows on Compact Surfaces of Genus g > 1 in the Class of Functions Analytic in Momenta.- 4.2. The Case of a Sphere and a Torus.- 4.3. The Properties of Integrable Geodesic Flows on the Sphere.- 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.