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# Integrability and Nonintegrability in Geometry and Mechanics

## byA.T. Fomenko

### Paperback | October 1, 2011

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Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. 1hen one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Oad in Crane Feathers' in R. Brown 'The point of a Pin' . . 1111 Oulik'. n. . Chi" .. . ~ Mm~ Mu,d. ", Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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Title:Integrability and Nonintegrability in Geometry and MechanicsFormat:PaperbackPublished:October 1, 2011Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401078807

ISBN - 13:9789401078801

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Table of Contents

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.

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