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Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds

Classical and Quantum Aspects
BuchKartoniert, Paperback
559 Seiten
Englisch
Springer Netherlandserschienen am10.10.2012Softcover reprint of the original 1st ed. 1998
In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets.mehr
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Produkt

KlappentextIn recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets.
Details
ISBN/GTIN978-94-010-6096-7
ProduktartBuch
EinbandartKartoniert, Paperback
Erscheinungsjahr2012
Erscheinungsdatum10.10.2012
AuflageSoftcover reprint of the original 1st ed. 1998
ReiheMathematics and Its Applications
Seiten559 Seiten
SpracheEnglisch
Gewicht890 g
Illustrationen559 p.
Artikel-Nr.30370920

Inhalt/Kritik

Inhaltsverzeichnis
1 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.mehr