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.

Inhalt

1. Some Equations of Classical Mechanics and Their Hamiltonian Properties.- §1. Classical Equations of Motion of a Three-Dimensional Rigid Body.- §2. Symplectic Manifolds.- §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.- 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.- §2. Multidimensional Integrable Systems. Classification of the Surgery on Liouville Tori in the Neighbourhood of Bifurcation Diagrams.- §3. The Properties of Decomposition of Constant-Energy Surfaces of Integrable Systems into the Sum of Simplest Manifolds.- 3. Some General Principles of Integration of Hamiltonian Systems of Differential Equations.- §1. Noncommutative Integration Method.- §2. The General Properties of Invariant Submanifolds of Hamiltonian Systems.- §3. Systems Completely Integrable in the Noncommutative Sense Are Often Completely Liouville-Integrable in the Conventional Sense.- §4. Liouville Integrability on Complex Symplectic Manifolds.- 4. Integration of Concrete Hamiltonian Systems in Geometry and Mechanics. Methods and Applications.- §1. Lie Algebras and Mechanics.- §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.- §3. Euler Equations on the Lie Algebra so(4).- §4. Duplication of Integrable Analogues of the Euler Equationsby Means of Associative Algebra with Poincaré Duality.- §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.- §2. Topological Obstacles for Complete Integrability.- §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.- 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.