Nonlinear dynamical systems of mathematical physics :: spectral and symplectic integrability analysis /
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 i...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
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Singapore ; Hackensack, NJ :
World Scientific,
©2011.
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| Online-Zugang: | Volltext |
| Zusammenfassung: | 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. |
| Beschreibung: | 1 online resource (xix, 542 pages) |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9789814327169 9814327166 1283234793 9781283234795 9786613234797 6613234796 |
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| 505 | 0 | |a 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. | |
| 520 | |a 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. | ||
| 546 | |a English. | ||
| 650 | 0 | |a Differentiable dynamical systems. |0 http://id.loc.gov/authorities/subjects/sh85037882 | |
| 650 | 0 | |a Nonlinear theories. |0 http://id.loc.gov/authorities/subjects/sh85092332 | |
| 650 | 0 | |a Symplectic geometry. |0 http://id.loc.gov/authorities/subjects/sh2002004420 | |
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| 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. |
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| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Electronic eBook |
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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. 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| illustrated | Not Illustrated |
| indexdate | 2024-11-27T13:18:01Z |
| institution | BVB |
| isbn | 9789814327169 9814327166 1283234793 9781283234795 9786613234797 6613234796 |
| language | English |
| oclc_num | 754792913 |
| open_access_boolean | |
| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xix, 542 pages) |
| psigel | ZDB-4-EBA |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | World Scientific, |
| record_format | marc |
| spelling | Blackmore, Denis L. Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / Denis Blackmore, Anatoliy K. Prykarpatsky, Valeriy Hr. Samoylenko. Singapore ; Hackensack, NJ : World Scientific, ©2011. 1 online resource (xix, 542 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Includes bibliographical references and index. Print version record. 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. 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. English. Differentiable dynamical systems. http://id.loc.gov/authorities/subjects/sh85037882 Nonlinear theories. http://id.loc.gov/authorities/subjects/sh85092332 Symplectic geometry. http://id.loc.gov/authorities/subjects/sh2002004420 Spectrum analysis Mathematics. Dynamique différentiable. Théories non linéaires. Géométrie symplectique. SCIENCE Physics Mathematical & Computational. bisacsh Differentiable dynamical systems fast Nonlinear theories fast Symplectic geometry fast Prikarpatskiĭ, A. K. (Anatoliĭ Karolevich) https://id.oclc.org/worldcat/entity/E39PCjF8kcbBYMjPDYqv9qyTVy http://id.loc.gov/authorities/names/n91120122 Samoylenko, Valeriy Hr. has work: Nonlinear dynamical systems of mathematical physics (Text) https://id.oclc.org/worldcat/entity/E39PCGBryR6hCkXfQYGyxRRGwP https://id.oclc.org/worldcat/ontology/hasWork Print version: Blackmore, Denis L. Nonlinear dynamical systems of mathematical physics. Singapore ; Hackensack, NJ : World Scientific, ©2011 9789814327152 (DLC) 2010028336 (OCoLC)650019316 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=389627 Volltext |
| spellingShingle | Blackmore, Denis L. Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / 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. Differentiable dynamical systems. http://id.loc.gov/authorities/subjects/sh85037882 Nonlinear theories. http://id.loc.gov/authorities/subjects/sh85092332 Symplectic geometry. http://id.loc.gov/authorities/subjects/sh2002004420 Spectrum analysis Mathematics. Dynamique différentiable. Théories non linéaires. Géométrie symplectique. SCIENCE Physics Mathematical & Computational. bisacsh Differentiable dynamical systems fast Nonlinear theories fast Symplectic geometry fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85037882 http://id.loc.gov/authorities/subjects/sh85092332 http://id.loc.gov/authorities/subjects/sh2002004420 |
| title | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / |
| title_auth | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / |
| title_exact_search | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / |
| title_full | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / Denis Blackmore, Anatoliy K. Prykarpatsky, Valeriy Hr. Samoylenko. |
| title_fullStr | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / Denis Blackmore, Anatoliy K. Prykarpatsky, Valeriy Hr. Samoylenko. |
| title_full_unstemmed | Nonlinear dynamical systems of mathematical physics : spectral and symplectic integrability analysis / Denis Blackmore, Anatoliy K. Prykarpatsky, Valeriy Hr. Samoylenko. |
| title_short | Nonlinear dynamical systems of mathematical physics : |
| title_sort | nonlinear dynamical systems of mathematical physics spectral and symplectic integrability analysis |
| title_sub | spectral and symplectic integrability analysis / |
| topic | Differentiable dynamical systems. http://id.loc.gov/authorities/subjects/sh85037882 Nonlinear theories. http://id.loc.gov/authorities/subjects/sh85092332 Symplectic geometry. http://id.loc.gov/authorities/subjects/sh2002004420 Spectrum analysis Mathematics. Dynamique différentiable. Théories non linéaires. Géométrie symplectique. SCIENCE Physics Mathematical & Computational. bisacsh Differentiable dynamical systems fast Nonlinear theories fast Symplectic geometry fast |
| topic_facet | Differentiable dynamical systems. Nonlinear theories. Symplectic geometry. Spectrum analysis Mathematics. Dynamique différentiable. Théories non linéaires. Géométrie symplectique. SCIENCE Physics Mathematical & Computational. Differentiable dynamical systems Nonlinear theories Symplectic geometry |
| url | https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=389627 |
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