ISBN: 9780387950006

Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter. Books List_Books, [PU: Springer]

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ISBN: 9780387950006

A solid introduction to applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented such that graduates and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory, with many of the topics presented in a novel way, emphasising explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter., [PU: Springer]

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Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics (107), Band 107)

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2013, ISBN: 9780387950006

Springer, Taschenbuch, Auflage: 2nd ed. 1993. 3rd printing 2000, 548 Seiten, Publiziert: 2013-10-04T00:00:01Z, Produktgruppe: Book, Hersteller-Nr.: 10 black & white illustrations, 3.7 kg, Verkaufsrang: 1007, Algebra & Zahlentheorie, Naturwissenschaft & Mathematik, Fachbücher, Kategorien, Bücher, Analysis, Naturwissenschaften & Technik, Format: Illustriert, Springer, 2013

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2000, ISBN: 9780387950006

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ISBN: 9780387950006

Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie grou… More...

ISBN: 9780387950006

A solid introduction to applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented such that graduates and re… More...

Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics (107), Band 107)

*- Paperback*

2013

## ISBN: 9780387950006

Springer, Taschenbuch, Auflage: 2nd ed. 1993. 3rd printing 2000, 548 Seiten, Publiziert: 2013-10-04T00:00:01Z, Produktgruppe: Book, Hersteller-Nr.: 10 black & white illustrations, 3.7 kg,… More...

2000, ISBN: 9780387950006

Buch, Softcover, 2nd ed. 1993. 3rd printing 2000, [PU: Springer-Verlag New York Inc.], Springer-Verlag New York Inc., 2000

2000, ISBN: 9780387950006

[ED: 2], 2nd ed. 1993. 3rd printing 2000, Softcover, Buch, [PU: Springer-Verlag New York Inc.]

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Title: | |

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** Details of the book - Applications of Lie Groups to Differential Equations**

EAN (ISBN-13): 9780387950006

ISBN (ISBN-10): 0387950001

Hardcover

Paperback

Publishing year: 2007

Publisher: Springer-Verlag New York Inc.

548 Pages

Weight: 0,806 kg

Language: eng/Englisch

Book in our database since 2007-04-14T20:37:20-04:00 (New York)

Detail page last modified on 2021-10-14T05:26:45-04:00 (New York)

ISBN/EAN: 9780387950006

ISBN - alternate spelling:

0-387-95000-1, 978-0-387-95000-6

### Information from Publisher

Author: Peter J. Olver

Title: Graduate Texts in Mathematics; Applications of Lie Groups to Differential Equations

Publisher: Springer; Springer US

513 Pages

Publishing year: 2000-01-21

New York; NY; US

Weight: 1,680 kg

Language: English

52,38 € (DE)

53,85 € (AT)

75,07 CHF (CH)

POD

BC; Book; Hardcover, Softcover / Mathematik/Analysis; Mathematische Analysis, allgemein; Verstehen; CON_D035; B; Analysis; Group Theory and Generalizations; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Analysis; Group Theory and Generalizations; Calculus of Variations and Optimization; Systems Theory, Control; Mathematics and Statistics; Gruppen und Gruppentheorie; Variationsrechnung; Optimierung; Kybernetik und Systemtheorie

1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves and Connectedness.- 1.2. Lie Groups.- Lie Subgroups.- Local Lie Groups.- Local Transformation Groups.- Orbits.- 1.3. Vector Fields.- Flows.- Action on Functions.- Differentials.- Lie Brackets.- Tangent Spaces and Vectors Fields on Submanifolds.- Frobenius’ Theorem.- 1.4. Lie Algebras.- One-Parameter Subgroups.- Subalgebras.- The Exponential Map.- Lie Algebras of Local Lie Groups.- Structure Constants.- Commutator Tables.- Infinitesimal Group Actions.- 1.5. Differential Forms.- Pull-Back and Change of Coordinates.- Interior Products.- The Differential.- The de Rham Complex.- Lie Derivatives.- Homotopy Operators.- Integration and Stokes’ Theorem.- Notes.- Exercises.- 2 Symmetry Groups of Differential Equations.- 2.1. Symmetries of Algebraic Equations.- Invariant Subsets.- Invariant Functions.- Infinitesimal Invariance.- Local Invariance.- Invariants and Functional Dependence.- Methods for Constructing Invariants.- 2.2. Groups and Differential Equations.- 2.3. Prolongation.- Systems of Differential Equations.- Prolongation of Group Actions.- Invariance of Differential Equations.- Prolongation of Vector Fields.- Infinitesimal Invariance.- The Prolongation Formula.- Total Derivatives.- The General Prolongation Formula.- Properties of Prolonged Vector Fields.- Characteristics of Symmetries.- 2.4. Calculation of Symmetry Groups.- 2.5. Integration of Ordinary Differential Equations.- First Order Equations.- Higher Order Equations.- Differential Invariants.- Multi-parameter Symmetry Groups.- Solvable Groups.- Systems of Ordinary Differential Equations.- 2.6. Nondegeneracy Conditions for Differential Equations.- Local Solvability.- In variance Criteria.- The Cauchy—Kovalevskaya Theorem.- Characteristics.- Normal Systems.- Prolongation of 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.- The Adjoint Representation.- Classification of Subgroups and Subalgebras.- Classification of Group-Invariant Solutions.- 3.4. Quotient Manifolds.- Dimensional Analysis.- 3.5. Group-Invariant Prolongations and Reduction.- Extended Jet Bundles.- Differential Equations.- Group Actions.- The Invariant Jet Space.- Connection with the Quotient Manifold.- The Reduced Equation.- Local Coordinates.- Notes.- Exercises.- 4 Symmetry Groups and Conservation Laws.- 4.1. The Calculus of Variations.- The Variational Derivative.- Null Lagrangians and Divergences.- Invariance of the Euler Operator.- 4.2. Variational Symmetries.- Infinitesimal Criterion of Invariance.- Symmetries of the Euler—Lagrange Equations.- Reduction of Order.- 4.3. Conservation Laws.- Trivial Conservation Laws.- Characteristics of Conservation Laws.- 4.4. Noether’s Theorem.- Divergence Symmetries.- Notes.- Exercises.- 5 Generalized Symmetries.- 5.1. Generalized Symmetries of Differential Equations.- Differential Functions.- Generalized Vector Fields.- Evolutionary Vector Fields.- Equivalence and Trivial Symmetries.- Computation of Generalized Symmetries.- Group Transformations.- Symmetries and Prolongations.- The Lie Bracket.- Evolution Equations.- 5.2. Récursion Operators, Master Symmetries and Formal Symmetries.- Frechet Derivatives.- Lie Derivatives of Differential Operators.- Criteria for Recursion Operators.- The Korteweg—de Vries Equation.- Master Symmetries.- Pseudo-differential Operators.- Formal Symmetries.- 5.3. Generalized Symmetries and Conservation Laws.- Adjoints of Differential Operators.- Characteristics of Conservation Laws.- Variational Symmetries.- Group Transformations.- Noether’s Theorem.- Self-adjoint Linear Systems.- Action of Symmetries on Conservation Laws.- Abnormal Systems and Noether’s Second Theorem.- Formal Symmetries and Conservation Laws.- 5.4. The Variational Complex.- The D-Complex.- Vertical Forms.- Total Derivatives of Vertical Forms.- Functionals and Functional Forms.- The Variational Differential.- Higher Euler Operators.- The Total Homotopy Operator.- Notes.- Exercises.- 6 Finite-Dimensional Hamiltonian Systems.- 6.1. Poisson Brackets.- Hamiltonian Vector Fields.- The Structure Functions.- The Lie-Poisson Structure.- 6.2. Symplectic Structures and Foliations.- The Correspondence Between One-Forms and Vector Fields.- Rank of a Poisson Structure.- Symplectic Manifolds.- Maps Between Poisson Manifolds.- Poisson Submanifolds.- Darboux’ Theorem.- The Co-adjoint Representation.- 6.3. Symmetries, First Integrals and Reduction of Order.- First Integrals.- Hamiltonian Symmetry Groups.- Reduction of Order in Hamiltonian Systems.- Reduction Using Multi-parameter Groups.- Hamiltonian Transformation Groups.- The Momentum Map.- Notes.- Exercises.- 7 Hamiltonian Methods for Evolution Equations.- 7.1. Poisson Brackets.- The Jacobi Identity.- Functional Multi-vectors.- 7.2. Symmetries and Conservation Laws.- Distinguished Functionals.- Lie Brackets.- Conservation Laws.- 7.3. Bi-Hamiltonian Systems.- Recursion Operators.- Notes.- Exercises.- References.- Symbol Index.- Author Index.Symmetry methods have long been recognized to be of great importance for the study of the differential equations. This book provides a solid introduction to those applications of Lie groups to differential equations which have proved to be useful in practice. The computational methods are presented so that graduate students and researchers can readily learn to use them. Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.

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