Algebraic Integrability of Nonlinear Dynamical Systems on Manifolds: Classical and Quantum Aspects(443 Mathematics and Its Applications)

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. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi­ Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).

Table of Contents:
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.