Algebraic integrability of nonlinear dynamical systems on manifolds : classical and quantum aspects

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Algebraic integrability of nonlinear dynamical systems on manifolds : classical and quantum aspects / by Anatoliy K. Prykarpatsky and Ihor V. Mykytiuk

資料種別:

図書

出版情報:

Dordrecht : Kluwer Academic Press, c1998

形態:

553 p. ; 25 cm

シリーズ名:

Mathematics and its applications ; v. 443 <BA00044495>

著者名:

ISBN:

9780792350903 [0792350901]

書誌ID:

BA36739808

子書誌情報

フルテキスト

所蔵情報

他の版・巻

書誌詳細

注記:

Includes bibliographical references

主題:

Dynamics; Manifolds (Mathematics); Mathematical physics; Nonlinear theories; Quantum statistics

言語:

英語

目次情報:

Preface

Background Notations

Dynamical systems with homogeneous configuration spaces / 1：

Dynamical systems with symmetries

Poisson structures / 1.1：

Lie group actions on manifolds / 1.2：

Symplectic structures on coadjoint representation orbits / 1.3：

Hamiltonian group actions / 1.4：

Invariant Hamiltonian systems with homogeneous configuration spaces / 1.5：

The existence of a maximal involutive set of functions on the orbits of semi-simple elements of a semi-simple Lie algebra / 2：

The algebra of invariant polynomials on a semi-simple Lie algebra / 2.1：

Semi-simple element orbits / 2.2：

Maximal involutive sets of functions on semi-simple elements orbits / 2.3：

The integrability criterion and spherical pairs of Lie groups / 3：

Criterion of integrability / 3.1：

Spherical pairs of complex Lie groups / 3.2：

Interpolation property of spherical pairs of compact Lie groups / 4：

Properties of spherical pairs of compact Lie algebras / 4.1：

Points in general position / 4.2：

Spherical pairs of classical simple Lie groups / 5：

Preliminary remarks / 5.1：

Involutions of simple Lie algebras / 5.2：

Spherical pairs of classical Lie algebras / 5.3：

Classification of spherical pairs of the exceptional simple Lie algebras / 6：

Classification of spherical pairs of semi-simple Lie groups / 7：

Spherical pairs of semi-simple Lie algebras / 7.1：

Geometric quantization and integrable dynamical systems

Connections on line bundles

Line bundles

Equivalence classes of line bundles

Connections

The integrality condition

Hermitian structures

Equivalence classes of line bundles with connections / 1.6：

Holomorphic line bundles / 1.7：

Derivations / 1.8：

Tensor products, square roots and invariant Hermitian structures / 1.9：

Parallel transport on a line bundle with a connection / 1.10：

Parallel transport and derivations / 1.11：

Flat partial connections

Flat F-connections

Maps of line bundles

Line bundles of differential forms

Geometric quantization

Polarizations

Geometric quantization and reduction

Introduction

Hamiltonian reduction

Quantization / 4.3：

Examples: geometric quantization of the oscillator type Hamiltonian systems

Geometric quantization of the generalized n-dimensional harmonic oscillator

Geometric quantization of geodesic flow on a sphere

Geometric quantization of the multidimensional Kepler problem

Geometric quantization of the MIC-Kepler problem / 5.4：

Structures on manifolds and algebraic integrability of dynamical systems

Poisson structures and dynamical systems with symmetries

Preliminary notes

Poisson structures on manifolds

Casimir functions and involution

Casimir functions associated with classical expansions of semi-simple Lie algebras

Alder-Kostant-Symes and Mishchenko-Fomenko theorems

The reduction method and Poisson structures on dual spaces of semi-direct sums of Lie algebras

The mapping canonicity of symplectic structures

Momentum mapping

Poisson structures on dual spaces of semi-direct sums of Lie algebras

Canonical mappings / 2.4：

Nonlinear Neumann type dynamical systems as integrable flows on coadjoint orbits of Lie groups

The Neumann problem

The Lie-Poisson bracket associated with the ad-semidirect sum of Lie algebras

The canonical symplectic structure on T(S[superscript n-1]) and its diffeomorphisms / 3.3：

An involutive system of integrals for the Neumann dynamical system on the sphere S[superscript n-1] / 3.4：

Generalized Neumann-Bogoliubov dynamical systems / 3.5：

Abelian integrals, integrable dynamical systems, and their Lax type representations

The Neumann-Rosochatius-Bogoliubov Hamiltonian system

Conservation laws

Lax type representation

Dual momentum mappings and their applications

Preliminaries

Dualities

The Neumann-Rosochatius system

The Lie algebraic setting of Benney-Kaup dynamical systems and associated via Moser Neumann-Bogoliubov oscillatory flows

The Novikov-Lax finite-dimensional invariant reductions on nonlocal submanifolds / 6.1：

The Moser map and its associated dual moment maps into loop Lie algebras / 6.3：

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

The Moser type of finite-dimensional reduction of a Boussinesq hydrodynamic system and its Lie-algebraic integrability / 7.2：

The Neumann type of oscillatory super-Hamiltonian systems on the sphere S[superscript N] and their Lie algebraic super-integrability / 7.3：

Lax-type of flows on Grassmann manifolds and dual momentum mappings / 8：

Symplectic structures on loop Grassmann manifolds / 8.1：

An intrinsic loop Grassmannian structure and dual momentum mappings / 8.3：

On the geometric structure of integrable flows in Grassmann manifolds / 9：

Centrally extended symplectic structures and integrable flows on the loop Grassmann manifolds / 9.1：

Algebraic methods of quantum statistical mechanics and their applications

Current algebra representation formalism in nonrelativistic quantum mechanics

The current algebra in nonrelativistic quantum mechanics

Current algebra representations

Bogoliubov-Araki generating functional

Lie current algebra, Hamiltonian operator, and Bogoliubov functional equations

Hamiltonian operator

Gibbs states and the Kubo-Martin-Schwinger conditions

Stable states and the KMS condition

Functional-operator representations of the current Lie algebra

The Bogoliubov-Bloch functional equation / 2.5：

The reconstruction via Araki of the Hamiltonian operator and the Bogoluibov functional equation / 2.6：

Functional-operator solutions of the Bogoliubov functional equations / 2.7：

A generalized Virasoro algebra / 2.8：

The secondary quantization method and the spectrum of quantum excitations of a nonlinear Schrodinger type dynamical system

Preliminary notions

The second quantization representation

A generalized nonlinear Schrodinger type quantum dynamical system

The quantum inverse spectral transform method

The scattering operator

Eigenvalue states of the nonlinear Schrodinger type model / 3.6：

Quantum excitations of a bose gas with a positive momentum / 3.7：

Unitary representations of the generalized Virasoro algebra

Verma modules over the generalized Virasoro current algebra

Unitary irreducible modules with highest weight

Algebraic and differential geometric aspects of the integrability of nonlinear dynamical systems on infinite-dimensional functional manifolds

The current Lie algebra on S[superscript 1] and its functional representations

Basic notations

Associated cohomology complexes and their properties

Differential geometry analysis on real jet manifolds

Differential geometry analysis on jet supermanifolds

Euler variational derivative, external differential and tensors on infinite-dimensional functional spaces

Implectic operators and Poisson structures

Dynamical systems and bi-Hamiltonicity

The equivalence of dynamical systems as the Backlund transformation

The current Lie algebra on a cycle as a symmetry subalgebra of compatibly bi-Hamiltonian nonlinear dynamical systems on an axis

The Hamiltonicity of nonlinear dynamical systems on infinite-dimensional functional manifolds

A Lie-algebraic algorithm for investigating integrability

The gradient holonomic algorithm and Lax type representation

An adjoint Lax type equation and conservation laws

Lax type representation: differential algebraic approach

Lax type representation: geometric approach

Lagrangian and Hamiltonian formalisms for reduced infinite-dimensional dynamical systems with symmetries

General setting

Lagrangian reduction

Symplectic analysis and Hamiltonian fields

Discrete dynamical systems. One generalization

Non-isospectrally integrable dynamical systems: the generalized asymptotic structure of conservation laws

A nonstandard reduction problem

Lagrangian and Hamiltonian analysis of dynamical systems on functional manifolds. The Poisson-Dirac bracket / 3.8：

Conclusions / 3.9：

The algebraic structure of the gradient-holonomic algorithm for Lax type integrable nonlinear dynamical systems

Algebraic structure of the Lax type integrable dynamical system

The periodic problem and canonical variational relationships

The spectral gradient structure of Lax integrable many-dimensional nonlinear dynamical systems on operator manifolds / 4.4：

The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra

An effective Maurer-Cartan one-form construction

General structure of integrable one-forms augmenting the two-forms associated with a closed ideal in the Grassmann algebra

Cartan's invariant geometric object structure of the gradient-holonomic algorithm for Lax integrable nonlinear dynamical systems in partial derivatives

A loop algebra and the Yang-Baxter structure / 5.5：

The transfer matrix properties / 5.6：

The algebraic structure of the gradient-holonomic algorithm for the Lax-type nonlinear dynamical systems: the reduction via Dirac and the canonical quantization procedure

The generalized R-structure hierarchy

The Dirac quantization procedure for Moser induced finite-dimensional Neumann type Hamiltonian systems

The scalar and operator integrable Hamiltonian systems via the algebraic gradient-holonomic algorithm / 6.4：

Concluding remarks / 6.5：

Hamiltonian structures of hydrodynamical Benny type dynamical systems and their associated Boltzmann-Vlasov kinetic equations on an axis

The Boltzmann equation and an associated moment problem

A nonlinear completely integrable Schrodinger type dynamical system approximation

The complete integrability of a Benney type hydrodynamical system associated with a Boltzmann-Vlasov equation / 7.4：

Conclusion / 7.5：

Appendix

Basic definitions, examples / .1：

The tangent Lie algebra / .2：

Lie subgroups / .3：

Lie algebras / .4：

Cartan subalgebras / .5：

Semi-simple complex Lie algebras / .6：

References

Preface

Background Notations

Dynamical systems with homogeneous configuration spaces / 1：

Dynamical systems with symmetries

Poisson structures / 1.1：

Lie group actions on manifolds / 1.2：

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