Beschreibung
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.
Autorenporträt
Inhaltsangabe1. 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
