Applications of Lie Groups to Differential Equations / by Peter J. Olver
(Graduate Texts in Mathematics ; 107)
| データ種別 | 電子ブック |
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| 出版者 | New York, NY : Springer US |
| 出版年 | 1986 |
| 本文言語 | 英語 |
| 大きさ | online resource |
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| 内容注記 | 1 Introduction to Lie Groups 1.1. Manifolds 1.2. Lie Groups 1.3. Vector Fields 1.4. Lie Algebras 1.5. Differential Forms Notes Exercises 2 Symmetry Groups of Differential Equations 2.1. Symmetries of Algebraic Equations 2.2. Groups and Differential Equations 2.3. Prolongation 2.4. Calculation of Symmetry Groups 2.5. Integration of Ordinary Differential Equations 2.6. Nondegeneracy Conditions for Differential Equations Notes Exercises 3 Group-Invariant Solutions 3.1. Construction of Group-Invariant Solutions 3.2. Examples of Group-Invariant Solutions 3.3. Classification of Group-Invariant Solutions 3.4. Quotient Manifolds 3.5. Group-Invariant Prolongations and Reduction Notes Exercises 4 Symmetry Groups and Conservation Laws 4.1. The Calculus of Variations 4.2. Variational Symmetries 4.3. Conservation Laws 4.4. Noether’s Theorem Notes Exercises 5 Generalized Symmetries 5.1. Generalized Symmetries of Differential Equations 5.2. Recursion Operators 5.3. Generalized Symmetries and Conservation Laws 5.4. The Variational Complex Notes Exercises 6 Finite-Dimensional Hamiltonian Systems 6.1. Poisson Brackets 6.2. Symplectic Structures and Foliations 6.3. Symmetries, First Integrals and Reduction of Order Notes Exercises 7 Hamiltonian Methods for Evolution Equations 7.1. Poisson Brackets 7.2. Symmetries and Conservation Laws 7.3. Bi-Hamiltonian Systems Notes Exercises References Symbol Index Author Index |
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| 一般注記 | This book is devoted to explaining a wide range of applications of con tinuous symmetry groups to physically important systems of differential equations. Emphasis is placed on significant applications of group-theoretic methods, organized so that the applied reader can readily learn the basic computational techniques required for genuine physical problems. The first chapter collects together (but does not prove) those aspects of Lie group theory which are of importance to differential equations. Applications covered in the body of the book include calculation of symmetry groups of differential equations, integration of ordinary differential equations, including special techniques for Euler-Lagrange equations or Hamiltonian systems, differential invariants and construction of equations with pre scribed symmetry groups, group-invariant solutions of partial differential equations, dimensional analysis, and the connections between conservation laws and symmetry groups. Generalizations of the basic symmetry group concept, and applications to conservation laws, integrability conditions, completely integrable systems and soliton equations, and bi-Hamiltonian systems are covered in detail. The exposition is reasonably self-contained, and supplemented by numerous examples of direct physical importance, chosen from classical mechanics, fluid mechanics, elasticity and other applied areas |
| 著者標目 | *Olver, Peter J. author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Topological groups LCSH:Lie groups FREE:Mathematics FREE:Topological Groups, Lie Groups |
| 分 類 | DC23:512.55 DC23:512.482 |
| 巻冊次 | ISBN:9781468402742 
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| ISBN | 9781468402742 |
| URL | http://dx.doi.org/10.1007/978-1-4684-0274-2 |
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