| Description |
1 online resource (xix, 542 pages) |
| Physical Medium |
polychrome |
| Description |
text file |
| Bibliography |
Includes bibliographical references and index. |
| Contents |
1. General properties of nonlinear dynamical systems. 1.1. Finite-dimensional dynamical systems. 1.2. Poissonian and symplectic structures on manifolds -- 2. Nonlinear dynamical systems with symmetry. 2.1. 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 method. 2.6. The geometric structure of non-abelian Yang-Mills gauge field equations via the reduction method. 2.7. Classical and quantum integrability -- 3. Integrability by quadratures. 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 -- 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. Backlund transformations. 4.5. Properties of solutions of some infinite sequences of dynamical systems. 4.6. Integro-differential systems -- 5. Integrability : The gradient-holonomic algorithm. 5.1. The Lax representation. 5.2. Recursive operators and conserved quantities. 5.3. Existence criteria for a Lax representation. 5.4. The current Lie algebra on a cycle : A symmetry subalgebra of compatible bi-Hamiltonian nonlinear dynamical systems -- 6. Algebraic, differential and geometric aspects of integrability. 6.1. A non-isospectrally Lax integrable KdV dynamical system. 6.2. Algebraic structure of the gradient-holonomic algorithm for Lax integrable systems. |
| 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, novel treatments of classical integration by quadratures, Lie-algebraic characterizations of integrability, and recent results on tensor Poisson structures. Of particular note is the development via spectral reduction of a generalized de Rham-Hodge theory, related to Delsarte-Lions operators, leading to new Chern type classes useful for integrability analysis. Also included are elements of quantum mathematics along with applications to Whitham systems, gauge theories, hadronic string models, and a supplement on fundamental differential-geometric concepts making this volume essentially self-contained. This book is ideal as a reference and guide to new directions in research for advanced students and researchers interested in the modern theory and applications of integrable (especially infinite-dimensional) dynamical systems. |
| Local Note |
eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America |
| Subject |
Differentiable dynamical systems.
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Differentiable dynamical systems. |
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Nonlinear theories.
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Nonlinear theories. |
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Symplectic geometry.
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Symplectic geometry. |
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Spectrum analysis -- Mathematics.
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Spectrum analysis. |
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Mathematics. |
| Genre/Form |
Electronic books.
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Electronic books.
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| Added Author |
Prikarpatskiĭ, A. K. (Anatoliĭ Karolevich)
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Samoylenko, Valeriy Hr.
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| Other Form: |
Print version: Blackmore, Denis L. Nonlinear dynamical systems of mathematical physics. Singapore ; Hackensack, NJ : World Scientific, ©2011 9789814327152 (DLC) 2010028336 (OCoLC)650019316 |
| ISBN |
9789814327169 (electronic book) |
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9814327166 (electronic book) |
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9789814327152 |
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9814327158 |
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