This volume contains five surveys on dynamical systems. The first one deals with nonholonomic mechanics and gives an updated and systematic treatment of the geometry of distributions and ofvariational problems with noninteg-rable constraints. The modern language of differential geometry used throughout the survey allows for a clear and tmified exposition of the earlier work on nonholonomic problems. There is a detailed discussion of the dynamical properties of the nonholonomic geodesic flow and of various related concepts, such as nonholonomic exponential mapping, nonholonomic sphere, etc. Other surveys treatvarious aspects of integrable Hamiltonian systems, with an emphasis on Lie-algebraic constructions. Among the topics covered are: the generalized Calogero-Moser systems based on root systems of simple Lie algebras, a general r-matrix scheme for constructing integrable systems and Lax pairs, links, with finite-gap integration theory, topological aspects of integrable systerms, integrable tops, etc. One of the surveys gives a thorough analysis of a family of quantum integrable systems(Toda lattices)using the machinery of representation theory. Readers will find all the new differential geometric and Lie-algebraic rnethods which are currently used in the theory ofintegrable systems in this book. It will be indispensable to graduate students and researchers in mathematics and theoretical physics.
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目录
Ⅰ. Nonholonomic Dynamical Systems. Geometry Introduction Chapter 1 Geometry of Distributions §1. Distributions and Related Objects 1.1. Distributions and Differential Systems 1.2. Frobenius Theorem and the Flag of a Distribution 1.3. Codistributions and Pfaffian Systems 1.4. Regular Distributions 1.5. Distributions Invariant with Respect to Group Actions and Some Canonical Examples 1.6. Connections as Distributions 1.7. A Classification of Left Invariant Contact Structures on Three-Dimensional Lie Groups §2. Generic Distributions and Sets of Vector Fields，and Degeneracies of Small Codimension. Nilpotentization and Classification Problem 2.1. Generic Distributions 2.2. Normal Forms of Jets of Basic Vector Fields of a Generic Distribution 2.3. Degeneracies of Small Codimension 2.4. Generic Sets of Vector Fields 2.5. Small Codimension Degeneracies of Sets of Vector Fields 2.6. Projection Map Associated with a Distribution 2.7. Classifrcaton of Regular Distributions 2.8. Nilpotentization and Nilpotent Calculus Chapter 2 Basic Theory of Nonholonomic Riemannian Manifolds §1. General Nonholonomic Variational Problem and the Geodesic Flow on Nonholonomic Riemannian Manifolds 1.1. Rashevsky-Chow Theorem and Nonholonomic Riemannian Metrics (Carnot-Carathkodory Metrics) 1.2. Two-Point Problem and the Hopf-Rinow Theorem 1.3. The Cauchy Problem and the Nonholonomic Geodesic Flow 1.4. The Euler-Lagrange Equations in Invariant Form and in the Orthogonal Moving Frame and Nonholonomic Geodesics 1.5. The Standard Form of Equations of Nonholonomic Geodesics for Generic Distributions 1.6. Nonholonomic Exponential Mapping and the Wave Front 1.7. The Action Functional §2. Estimates of the Accessibility Set 2.1. The Parallelotope Theorem 2.2. Polysystems and Finslerian Metrics 2.3. Theorem on the Leading Term 2.4. Estimates of Generic Nonholonomic Metrics on Compact Manifolds 2.5. Hausdorff Dimension of Nonholonomic Riemannian Manifolds 2.6. The Nonholonomic Ball in the Heisenberg Group as the Limit of Powers of a Riemannian Ball Chapter 3 Nonholonomic Variational Problems on Three-Dimensional Lie Groups §1. The Nonholonomic &-Sphere and the Wave Front 1.1. Reduction of the Nonholonomic Geodesic Flow 1.2. Metric Tensors on Three-Dimensional Nonholonomic Lie Algebras 1.3. Structure Constants of Three-Dimensional Nonholonomic Lie Algebras 1.4. Normal Forms of Equations of Nonholonomic Geodesics on Three-Dimensional Lie Groups 1.5. The Standard Form of Equations of Nonholonomic Geodesics for Generic Distributions 1.6. Wave Front of Nonholonomic Geodesic Flow，Nonholonomic ε-Sphere and their Singularities 1.7. Metric Structure of the Sphere SL §2. Nonholonomic Geodesic Flow on Three-Dimensional Lie Groups 2.1. The Monodromy Maps 2.2. Nonholonomic Geodesic Flow on SO(3) 2.3. NG-Flow on Compact Homogeneous Spaces of the Heisenberg Group 2.4. Nonholonomic Geodesic Flows on Compact Homogeneous Spaces of SL，R 2.5. Nonholonomic Geodesic Flow on Some Special Multidimensional Nilmanifolds References Additional Bibliographical Notes Ⅱ. Integrable Systems Ⅱ Introduction Chapter 1 Integrable Systems and Finite-Dimensional Lie Algebras (M.A. Olshanetsky. A.M. Perelomov) §1. Hamiltonian Systems on Coadjoint Orbits of Lie Groups §2. The Moment Map §3. The Projection Method §4. Description of the Calogero-Sutherland Type Systems §5. The Toda Lattice §6. Lax Representation. Proof of Complete Integrability §7. Explicit Integration of the Equations of Motion A) Systems of Calogero Type B) Systems of Sutherland Type C) Systems with Two Types of Particles D) The Nonperiodic Toda Lattice §8. Bibliographical Notes References Chapter 2. Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems (A.G. Reyman. M. A. Semenov-Tian-Shansky) Introduction §1. Poisson Manifolds §2. The R-Matrix Method and the Main Theorem 2.1. The Involutivity Theorem 2.2. Factorization Theorem 2.3. Classical Yang-Baxter Identity and the General Theory of Classical R-Matrices 2.4. Examples 2.5. Bibliographical Notes §3. Gradings and Orbits. Toda Lattices 3.1. Graded Algebras 3.2. First Examples: Toda Lattices 3.3. Solutions and Scattering for Toda Lattices 3.4. Bibliographical Notes §4. Affine Lie Algebras and Lax Equations with a Spectral Parameter 4.1. Construction of Loop Algebras 4.2. Hierarchies of Poisson Structures for Lax Equations with a Spectral Parameter 4.3. Structure Theory of Affine Lie Algebras 4.4. Periodic Toda Lattices 4.5. Multi-pole Lax Equations 4.6. Bibliographical Notes §5. Hamiltonian Reduction and Orbits of Semi-direct Products 5.1. Hamiltonian Reduction 5.2. Examples. Magnetic Cotangent Bundles 5.3. Orbits of Semi-direct Products 5.4. Nonabelian Toda Lattices 5.5. Bibliographical Notes §6. Lax Representations of Multi-dimensional Tops 6.1. Kinematics of the n-dimensional Rigid Body 6.2. Integrable Top-Like Systems 6.3. Bibliographical Notes §7. Integrable Multi-dimensional Tops and Related Systems 7.1. (G，K) = (GL(n，R)，SO(n))：Multidimensional Tops in Quadratic Potentials. The Neumann System. Multidimensional Rigid Body in Ideal Fluid：The Clebsch Case 7.2. (G，K) = (SO(p，q)，SO(p)xSO(q))：Interacting Tops and Kowalewski Tops. p-dimensional Top Interacting with a Rotator. p-dimensional Lagrange's Heavy Top. Generalized Kowalewski Tops. Three-dimensional Tops Associated with the Lie Algebras eo(4，4) and so(4，3) 7.3. The Lie Algebra G2 an Exotic SO(4)-Top 7.4. The Anharmonic Oscillator，the Garnier System and Integrable Quartic Potentials 7.5. Bibliographical Notes §8. The Riemann Problem and Linearization of Lax Equations 8.1. The Riemann Problem 8.2. Spectral Data and Dynamics 8.3. Reconstruction of L(λ) from the Spectral Data 8.4. Bibliographical Notes §9. Completeness of the Integrals of Motion 9.1. The Case of *(gI(n，C)) 9.2. Complete Integrability for Other Lie Algebras 9.3. Bibliographical Notes §10. The Baker-Akhiezer Functions 10.1. Solution of the Matrix Riemann Problem 10.2. Explicit Formulae for the Baker-Akhiezer Functions 10.3. Example：Algebraic Geometry of the Kowalewski Top 10.4. Bibliographical Notes §11. Equations with an Elliptic Spectral Parameter 11.1. Elliptic Decomposition 11.2. Lax Equations with an Elliptic Parameter 11.3. Multi-pole Lax Equations 11.4. Riemann Problem，Cousin Problem and Spectral Data 11.5. Real Forms and Reduction 11.6. Examples 11.7. Hierarchies of Poisson Brackets 11.8. Bibliographical Notes §12. Classical R-Matrices. Poisson Lie Groups and Difference Lax Equations 12.1. Poisson Lie Groups 12.2. Duality Theory for Poisson Lie Groups 12.3. Poisson Reduction，Dressing Transformations and Symplectic Leaves of Poisson Lie Groups 12.4. Lax Equations on Lie Groups 12.5. Poisson Structure on Loop Groups and Applications 12.6. Additional Comments 12.7. Bibliographical Notes References Chapter 3. Quantization of Open Toda Lattices (M.A. Semenov-Tian-Shansky) Introduction §1. Reduction of Quantum Bundles 1.1. Lagrangian Polarizations 1.2. Reduction of Quantum Bundles 1.3. The Quantum Version of Kostant's Commutativity Theorem 1.4. The Generalized Kostant Theorem and Reduction of Quantum Bundles 1.5. Quantization. The Duflo Homomorphism §2. Quantum Toda Lattices 2.1. Semisimple Lie Groups and Lie Algebras. Notation 2.2. Toda Lattices：a Geometric Description 2.3. Toda Lattices：Geometric Quantization §3. Spectral Theory of the Quantum Toda Lattice 3.1. Representations of the Principal Series and Whittaker Functions 3.2. Analytic Properties of the Whittaker Functions 3.3. The Spectral Decomposition Theorem 3.4. Degenerate Toda Lattices 3.5. The Wave Packets and their Properties 3.6. Scattering Theory 3.7. Sketch of a Proof of the Main Theorem 3.8. Induction Over Rank Bibliographical Notes References Ⅲ. Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras Chapter 1 Geometry and Topology of Hamiltonian Systems §1. Symplectic Geometry 1.1. Symplectic Manifolds 1.2. Embeddings of Symplectic Manifolds 1.3. Symplectic Geometry of the Coadjoint Representation 1.4. Poisson Structures on Lie Algebras 1.5. Euler Equations 1.6. Euler Equations Arising in Problems of Mathematical Physics §2. Some Classical Mechanisms of Integrability 2.1. The Hamilton-Jacobi Equation 2.2. Integration of the Equations of Motion According to Liouville and Stackel 2.3. Lie's Theorem 2.4. Liouville's Theorem §3. Non-commutative Integration According to Liouville 3.1. Non-commutative Lie Algebras of Integrals 3.2. Non-commutative Liouville Theorem 3.3. Interrelationships of Systems with Commutative and Non-commutative Symmetries 3.4. Local Equivalence of Commutative and Non-commutative Integration §4. The Geometry of the Moment Map 4.1. The Moment Map 4.2. Convexity Properties of the Moment Map 4.3. Multiplicity-free Representations §5. The Topology of Surfaces of Constant Energy in Completely Integrable Hamiltonian Systems 5.1. The Multi-dimensional Case. Classification of Surgery of Liouville Tori 5.2. The Four-dimensional Case 5.3. The Case of Four-dimensional Rigid Body Chapter 2 The Algebra of Hamiltonian Systems §1. Representations of Lie Groups and Dynamical Systems 1.1. Symplectic Structures Associated with Representations 1.2. Sectional Operators 1.3. Integrals of Euler Equations. Shift of the Argument 1.4. Sectional Operators for Symmetric Spaces 1.5. Complex Semisimple Series of Sectional Operators. 1.6. Compact and Normal Series of Sectional Operators 1.7. Sectional Operators for the Lie Algebra of the Group of Euclidean Motions 1.8. Sectional Operators for the Lie Algebra Ω(g) 1.9. Bi-Hamiltonian Properties of Euler Equations on Semisimple Lie Algebras §2. Methods of Constructing Functions in Involution 2.1. Inductive Construction of Integrable Dynamical Systems on Coadjoint Orbits (Chains of Subalgebras) 2.2. Representations of Lie Groups and Involutive Families of Functions 2.3. Involutive Families of Functions on Semidirect Sums 2.4. The Method of Tensor Extensions of Lie Algebras §3. Completely Integrable Euler Equations on Lie Algebras 3.1. Euler Equations on Semisimple Lie Algebras 3.2. Euler Equations on Solvable Lie Algebras 3.3. Euler Equations on Non-solvable Lie Algebras with a Non-trivial Radical 3.4. Integrable Systems and Symmetric Spaces 3.5. Theorem on the Completeness of Shifted Invariants Appendix References
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