Differential Equations: Their Solution Using Symmetries

Description

English transl. of

Stephani, Hans. 1994. Differentialgleichungen: Symmetrien und Lösungsmethoden. Heidelberg: Spektrum Akad. Verl.

cited by physicist

• Starrett, John. “Solving Differential Equations by Symmetry Groups.” The American Mathematical Monthly 114, no. 9 (2007): 778–92.

Stephani read Lie's original papers in German, and he considers nothing in his book original but Lie speaking.

ch. 10 (DjVu pp. 102ff.):

1. "10.1 The corresponding linear partial differential equation of first order" (how systems of 2nd order ODEs relate to PDEs).
2. §10.2: Kepler problem

DjVu pp. 40-42 (pp. 31-33) gives a concise, elegant proof for why there are no more than 8 symmetries of a 2nd order DE.

Appendix B contains solutions to the more difficult exercises.

MR1041800

This book is an introduction to the symmetry analysis of differential equations and its use for finding exact solutions. It consists of two parts. ODEs are considered in the first part. The Lie point symmetries are analyzed for equations of 1st, 2nd and higher orders, and also for the systems of equations of 1st and 2nd order. Furthermore, symmetries more general than point, contact and dynamical symmetries are studied. PDEs are considered in the second part. Point Lie symmetries are explored here as well as contact symmetries. Not only standard graduate material is discussed but also more special and modern problems: how to find dynamical symmetries for systems possessing a Lagrangian, symmetries and the separability of the Hamilton-Jacobi equation, Lie-Bäcklund symmetries and conservation laws, differential equations and symmetries in the language of forms. Many examples are discussed, and the book includes more than 100 exercises.
Thanks to its methodical and scientific advantages, this book is interesting not only for beginners, but also for researchers. Reviewed by L. M. Berkovich

In many branches of physics, mathematics, and engineering, solving a problem means solving a set of ordinary or partial differential equations. Nearly all methods of constructing closed form solutions rely on symmetries. The emphasis in this text is on how to find and use the symmetries; this is supported by many examples and more than 100 exercises. This book will form an introduction accessible to beginning graduate students in physics, applied mathematics, and engineering. Advanced graduate students and researchers in these disciplines will find the book a valuable reference.

**

."..as an account of classical general relativity, this well produced and excellently translated book has many virtues." Physics Bulletin "A nice introduction to the theory and practice of finding and using symmetries to solve differential equations." American Mathematical Monthly ."..Stephani's book does a good job of motivating the study of Lie group methods for differential equations from an elementary standpoint." SIAM Reviews "The author, who has an easy-to-read practical style, consistently keeps the emphasis on applications. Thus, most physicists will be able to get useful information about ordinary differential equations (ODE's) and partial differential equations (PDE's), without being bogged down in cumbersome mathematical formalism....well worth reading if one is at all interested in sophisticated and powerful symmetry techniques for handling differential equations and if one wishes to have the most straightforward approach to the topic." D. E. Vincent, Physics in Canada ."..Stephani...has built a book that tries to guide its readers toward a sure knowledge of this very important tool for finding solutions of (nonlinear) differential equations. In the early sections, the derivations presented are the most clear and detailed ones that this writer has ever seen....Students new to this area will find reading or studying the current book an altogether enjoyable occupation, without any of the intimidation that sometimes is caused by similar books." J.D. Finley, Foundations of Physics ..".as an account of classical general relativity, this well produced and excellently translated book has many virtues." Physics Bulletin ..".Stephani's book does a good job of motivating the study of Lie group methods for differential equations from an elementary standpoint." SIAM Reviews ..".Stephani...has built a book that tries to guide its readers toward a sure knowledge of this very important tool for finding solutions of (nonlinear) differential equations. In the early sections, the derivations presented are the most clear and detailed ones that this writer has ever seen....Students new to this area will find reading or studying the current book an altogether enjoyable occupation, without any of the intimidation that sometimes is caused by similar books." J.D. Finley, Foundations of Physics