Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / Denis Blackmore, Anatoliy K. Prykarpatsky, Valeriy Hr. Samoylenko.

Format:

Book

Author/Creator:

Blackmore, Denis L.

Contributor:

Prikarpatskiĭ, A. K. (Anatoliĭ Karolevich)

Samoylenko, Valeriy Hr.

Language:

English

Subjects (All):

Differentiable dynamical systems.

Nonlinear theories.

Symplectic geometry.

Spectrum analysis--Mathematics.

Spectrum analysis.

Physical Description:

1 online resource (563 p.)

Edition:

1st ed.

Place of Publication:

Singapore ; Hackensack, N.J. : World Scientific, c2011.

Language Note:

English

Summary:

This distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field - including some innovations by the authors themselves - that have not appeared in any other book. The exposition begins with an introduction to modern integrable dynamical systems theory, treating such topics as Liouville-Arnold and Mischenko-Fomenko integrability. This sets the stage for such topics as new formulations of the gradient-holonomic algorithm for Lax integrability,

Contents:

Preface; Contents; Chapter 1 General Properties of Nonlinear Dynamical Systems; 1.1 Finite-dimensional dynamical systems; 1.1.1 Invariant measure; 1.1.2 The Liouville condition; 1.1.3 The Poincaré theorem; 1.1.4 The Birkhoff-Khinchin theorem; 1.1.5 The Birkhoff-Khinchin theorem for discrete dynamical systems; 1.2 Poissonian and symplectic structures on manifolds; 1.2.1 Poisson brackets; 1.2.2 The Liouville theorem and Hamilton-Jacobi method; 1.2.3 Dirac reduction: Symplectic and Poissonian structures on submanifolds

Chapter 2 Geometric and Algebraic Properties of Nonlinear Dynamical Systems with Symmetry: Theory and Applications2.1 The Poisson structures and Lie group actions on manifolds: Introduction; 2.2 Lie group actions on Poisson manifolds and the orbit structure; 2.3 The canonical reduction method on symplectic spaces and related geometric structures on principal fiber bundles; 2.4 The form of reduced symplectic structures on cotangent spaces to Lie group manifolds and associated canonical connections

2.5 The geometric structure of abelian Yang-Mills type gauge field equations via the reduction method2.6 The geometric structure of non-abelian Yang-Mills gauge field equations via the reduction method; 2.7 Classical and quantum integrability; 2.7.1 The quantization scheme, observables and Poisson manifolds; 2.7.2 The Hopf and quantum algebras; 2.7.3 Integrable flows related to Hopf algebras and their Poissonian representations; 2.7.4 Casimir elements and their special properties; 2.7.5 Poisson co-algebras and their realizations

2.7.6 Casimir elements and the Heisenberg-Weil algebra related structures2.7.7 The Heisenberg-Weil co-algebra structure and related integrable flows; Chapter 3 Integrability by Quadratures of Hamiltonian and Picard-Fuchs Equations: Modern Differential-Geometric Aspects; 3.1 Introduction; 3.2 Preliminaries; 3.3 Integral submanifold embedding problem for an abelian Lie algebra of invariants; 3.4 Integral submanifold embedding problem for a nonabelian Lie algebra of invariants; 3.5 Examples; 3.6 Existence problem for a global set of invariants; 3.7 Additional examples

3.7.1 The Henon-Heiles system3.7.2 A truncated four-dimensional Fokker-Planck Hamiltonian system; Chapter 4 Infinite-dimensional Dynamical Systems; 4.1 Preliminary remarks; 4.2 Implectic operators and dynamical systems; 4.3 Symmetry properties and recursion operators; 4.4 Bäcklund transformations; 4.5 Properties of solutions of some infinite sequences of dynamical systems; 4.6 Integro-differential systems; Chapter 5 Integrability Criteria for Dynamical Systems: the Gradient-Holonomic Algorithm; 5.1 The Lax representation; 5.1.1 Generalized eigenvalue problem

5.1.2 Properties of the spectral problem

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

1-283-23479-3

9786613234797

981-4327-16-6

OCLC:

754792913