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OverviewThis book provides a detailed exposition of modern Lie-algebraic theory of integrable nonlinear dynamic systems on manifolds and its applications to mathematical physics, classical mechanics and hydrodynamics. The authors have developed a canonical geometric approach based on differential geometric considerations and spectral theory, which offers solutions to many quantization procedure problems. Much of the material is devoted to treating integrable systems via the gradient-holonomic approach devised by the authors, which can be very effectively applied. This volume is aimed at graduate-level students, researchers and mathematical physicists whose work involves differential geometry, ordinary differential equations, manifolds and cell complexes, topological groups and Lie groups. Full Product DetailsAuthor: A.K. Prykarpatsky , I.V. MykytiukPublisher: Springer Imprint: Springer Edition: 1998 ed. Volume: 443 Dimensions: Width: 17.00cm , Height: 3.10cm , Length: 24.40cm Weight: 2.120kg ISBN: 9780792350903ISBN 10: 0792350901 Pages: 559 Publication Date: 30 June 1998 Audience: College/higher education , Professional and scholarly , Undergraduate , Postgraduate, Research & Scholarly Format: Hardback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of Contents1 Dynamical systems with homogeneous configuration spaces.- 1 Dynamical systems with symmetries.- 2 The existence of a maximal involutive set of functions on the orbits of semi-simple elements of a semi-simple Lie algebra.- 3 The integrability criterion and spherical pairs of Lie groups.- 4 Interpolation property of spherical pairs of compact Lie groups.- 5 Spherical pairs of classical simple Lie groups.- 6 Classification of spherical pairs of the exceptional simple Lie algebras.- 7 Classification of spherical pairs of semi-simple Lie groups.- 2 Geometric quantization and integratble dynamical systems.- 1 Connections on line bundles.- 2 Flat partial connections.- 3 Geometric quantization.- 4.1 Introduction.- 5 Examples: geometric quantization of the oscillator type Hamiltonian systems.- 3 Structures on manifolds and algebraic integrability of dynamical systems.- 1 Poisson structures and dynamical systems with symmetries.- 2 The reduction method and Poisson structures on dual spaces of semi-direct sums of Lie algebras.- 3 Nonlinear Neumann type dynamical systems as integrable flows on coadjoint orbits of Lie groups.- 4 Abelian integrals, integrable dynamical systems, and their Lax type representations.- 5 Dual momentum mappings and their applications.- 6 The Lie algebraic setting of Benney-Kaup dynamical systems and associated via Moser Neumann-Bogoliubov oscillatory flows.- 7 The finite-dimensional Moser type of reduction of modified Boussinesq and super-Korteweg-de Vries Hamiltonian systems via the gradient-holonomic algorithm and dual moment maps.- 8 Lax-type of flows on Grassmann manifolds and dual momentum mappings.- 9 On the geometric structure of integrable flows in Grassmann manifolds.- 4 Algebraic methods of quantum statistical mechanics and their applications.-1 Current algebra representation formalism in nonrelativistic quantum mechanics.- 2 Lie current algebra, Hamiltonian operator, and Bogoliubov functional equations.- 3 The secondary quantization method and the spectrum of quantum excitations of a nonlinear Schrödinger type dynamical system.- 4 Unitary representations of the generalized Virasoro algebra.- 5 Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds.- 1 The current Lie algebra on S1 and its functional representations.- 2 The gradient holonomic algorithm and Lax type representation.- 3 Lagrangian and Hamiltonian formalisms for reduced infinite-dimensional dynamical systems with symmetries.- 4 The algebraic structure of the gradient-holonomic algorithm for Lax type integrable nonlinear dynamical systems.- 5 The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra.- 6 The algebraic structure of the gradient-holonomic algorithm for the Lax-type nonlinear dynamical systems: the reduction via Dirac and the canonical quantization procedure.- 7 Hamiltonian structures of hydrodynamical Benny type dynamical systems and their associated Boltzmann-Vlasov kinetic equations on an axis.- References.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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