Applications of Lie Groups to Differential Equations

1. Front Matter

   Pages i-xxvi

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2. Introduction to Lie Groups

   • Peter J. Olver

   Pages 1-76

3. Symmetry Groups of Differential Equations

   • Peter J. Olver

   Pages 77-185

4. Group-Invariant Solutions

   • Peter J. Olver

   Pages 186-245

5. Symmetry Groups and Conservation Laws

   • Peter J. Olver

   Pages 246-291

6. Generalized Symmetries

   • Peter J. Olver

   Pages 292-377

7. Finite-Dimensional Hamiltonian Systems

   • Peter J. Olver

   Pages 378-422

8. Hamiltonian Methods for Evolution Equations

   • Peter J. Olver

   Pages 423-474

9. Back Matter

   Pages 457-500

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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.

• Applications
• Equations
• Groups
• Lie
• group theory

• School of Mathematics, University of Minnesota, Minneapolis, USA

  Peter J. Olver

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