Applications of Lie Groups to Differential...

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Beschreibung

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.

Über den Autor

Peter Olver is Professor of Mathematics at University of Minnesota, Twin Cities. His research centers around Lie groups, differential equations, and various areas of applied mathematics. His previous books include Introduction to Partial Differential Equations (Springer, UTM, 2014), and Applications of Lie Groups to Differential Equations (Springer, GTM 107, 1993).

Chehrzad Shakiban is Professor of Mathematics at University of St. Thomas, St. Paul. Her interests include calculus of variations, computer vision, and innovative learning experiences that illustrate connections between mathematics and the arts.

Zusammenfassung

Inhaltsverzeichnis

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. Récursion Operators, Master Symmetries and Formal Symmetries.- 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.

Details

Erscheinungsjahr:	2000
Fachbereich:	Analysis
Genre:	Importe , Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Reihe:	Graduate Texts in Mathematics
Inhalt:	xxviii 513 S.
ISBN-13:	9780387950006
ISBN-10:	0387950001
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Olver, Peter J.
Auflage:	Second Edition 1993
Hersteller:	Humana Springer Springer US, New York, N.Y. Graduate Texts in Mathematics
Verantwortliche Person für die EU:	Springer Verlag GmbH, Tiergartenstr. 17, D-69121 Heidelberg, juergen.hartmann@springer.com
Maße:	235 x 155 x 30 mm
Von/Mit:	Peter J. Olver
Erscheinungsdatum:	21.01.2000
Gewicht:	0,82 kg