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Field name	Details
Dewey Class	512.55
Dewey Class	512.482
Title	Applications of Lie Groups to Differential Equations ([EBook] /) / by Peter J. Olver.
Author	Olver, Peter J.
Other name(s)	SpringerLink (Online service)
Publication	New York, NY : : Springer US, , 1986.
Physical Details	: online resource.
Series	Graduate texts in mathematics 0072-5285 ; ; 107
ISBN	9781468402742
Summary Note	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.:
Contents note	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.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-1-4684-0274-2
Links to Related Works	Subject References: Lie groups . Mathematics . Topological groups . Topological groups, Lie groups . Authors: Olver, Peter J. . Corporate Authors: SpringerLink (Online service) . Series: Graduate texts in mathematics . GTM . Classification: 512.482 . 512.55 .

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Applications of Lie Groups to Differential Equations

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