From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: Substitutions - How? Date: 15 Sep 1999 14:23:01 -0500 Newsgroups: sci.math Keywords: Applications of Lie theory for solving differential equations In article <37DF16AA.C48D07BB@cableol.co.uk> neenag@cableol.co.uk writes: > How does one know which substitution to use in a differential equation > or when solving an integration question? For example, in the following > differential equation: > > dy > -- + 3y = y^(2/3) > dx > > I would never be able to guess which substitution to use unless I was > told. (I was told using z = y^(1/3). On the first pass through a differential equations course, we usually learn a whole bag of tricks. Some tricks look familiar (like this power substitution), some seem similar to each other but not identical, and some may seem rather bizarre. A common feature is that, once you know the trick, there's not much left to do; however, the hard part is knowing which trick (and that there is a trick), which requires a good memory (and lots of practice). What they usually don't tell you is that there is, in fact, an approach which unifies all of these tricks (and can be used to derive them from "first principles", sort of). The basic idea is this. Associated with any ODE is a group of symmetry transformations. (One such transformation for the ODE you are looking at is the transformation (x,y) -> (x+c,y), where c is any constant. For some ODEs, this group is the trivial group.) You can exploit this symmetry group to help you choose new coordinates in which to write the ODE; in these new coordinates, the ODE should be fairly trivial. The results of this process may not look immediately like anything from your bag of tricks, and it sometimes takes some work to see how the relevant trick fits into this more complicated process. Of course I've left off some details. A fairly readable description is in Peter Olver's book "Applications of Lie Groups to Differential Equations", in section 2.5(?). The process is somewhat slower than just using the trick, though, so it's probably a good idea to just memorize the tricks (and, if possible, try to spot some simple patterns, like [as mentioned by someone else] "there's a power of y here, maybe I should substitute z = y^p"). Kevin. ============================================================================== From: "Bluesky" Subject: Lie's algebra and differential equations Date: Fri, 01 Oct 1999 11:05:42 GMT Newsgroups: sci.math It might be off-topic; but I do not know where to ask. Is anyone know any sources, papers, books on application of Lie's algrebra to (partial) differential equations ? So far I have only a simple book by Dresner. I want more in/formal discussion or examples on this tool. TIA. SilentNight ============================================================================== From: Robert Low Subject: Re: Lie's algebra and differential equations Date: Fri, 01 Oct 1999 16:09:53 +0100 Newsgroups: sci.math Bluesky wrote: > Is anyone know any sources, papers, books on > application of Lie's algrebra to (partial) > differential equations ? Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, Vol 107), Springer, by Peter J. Olver -- Rob. http://www.mis.coventry.ac.uk/~mtx014/ ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: Lie's algebra and differential equations Date: 04 Oct 1999 12:08:51 -0500 Newsgroups: sci.math In article <37F4CEC1.F07F5320@coventry.ac.uk> Robert Low writes: > Bluesky wrote: > > > Is anyone know any sources, papers, books on > > application of Lie's algrebra to (partial) > > differential equations ? > > Applications of Lie Groups to Differential Equations (Graduate Texts in > Mathematics, Vol 107), Springer, by Peter J. Olver Olver's book is readable, and present some useful material (especially regarding generalized and master symmetries and related things), but glosses over some important and subtle issues (and some of the computational methods are much more awkward than they need to be). His newer book on equivalence (I've forgotten the exact title) gives the tools for understanding these issues; however, a far better (in my opinion) source for these tools is Robert Gardner's "The Method of Equivalence and its Applications". You should probably read Bryant, Chern, Gardner, Goldschmidt, Griffiths, "Exterior Differential Systems", at least to familiarize yourself with "The Notion of an EDS" (a section in the first chapter); or a similarly titled book by Kichoon(?) Yang. The subtle issues are best dealt with by the Cartan-Kahler theorem, which may seem a large digression from the applications of Lie groups, but like I said, the issues are subtle. Then you can read Cartan. :-) Kevin. ============================================================================== From: "David Zachmann" Subject: Try Hans Stephani's Book Date: Wed, 6 Oct 1999 10:12:32 -0500 Newsgroups: sci.math Try: "Differential Equations : Their Solution Using Symmetries" Hans Stephani(Preface), Malcolm MacCallum (Editor) / Paperback / Published 1990 David Zachmann ------------------- Bluesky wrote in message news:aA0J3.570$a7.75301@nnrp.gol.com... [as above --djr] ============================================================================== From: Michael Jørgensen Subject: Re: Lie's Algebra and Differential Equations Date: Fri, 08 Oct 1999 09:39:31 +0200 Newsgroups: sci.math OK, perhaps you have enough already, but here is another book, one that I have read: "Symmetries and Differential Equations" by Bluman and Kumei (Applied Mathematical Sciences, Vol 81), 1989, 412 pages. This book starts of real slow with dimensional analysis, to get a feeling of the relevance of symmetries. Lots of algorithmic methods and easy-to-understand examples. The book covers both ordinary and partial DE's. Final chapters treat Noether's theorem and Lie-B=E4cklund symmetries. -Michael. ============================================================================== From: "Ed Severn" Subject: Re: Lie Group Theory Date: Fri, 21 May 1999 03:23:00 GMT Newsgroups: sci.math Jason, If you're interested in Lie groups in a practical sense, then try Peter Olver's "Applications of Lie Groups to Differential Equations", Springer GTM #107. Lie groups originated in the study of differential equations (What didn't?) and this book shows how to use them to solve ODE's that the undergraduate techniques don't solve. In fact, the usual handful of techniques that you learn in an undergraduate ODE course seem to work because of the structure of the underlying Lie group in which the DE "lives". I'm paraphrasing here since I know very little more than this. And other structural aspects of Lie groups lead to the well-known conservation laws of physics. So at least read the intro to Olver's book. Stephani has a similar book with the word "Symmetry" in its title. If you're interested in Lie groups for their own sake, though, then maybe someone else can recommend a good book. ...Ed Jason E Tedor wrote in message news:7i0fad$1g5$1@news.alaska.edu... > Looking for references to "good" introductory texts in Lie Group Theory at > the advanced undergraduate or graduate level. Any suggestions? > > Jason. ============================================================================== From: Chris Hillman Subject: Re: Lie Group Theory Date: Sat, 22 May 1999 12:59:04 -0700 Newsgroups: sci.math On Fri, 21 May 1999, Ed Severn wrote: > Jason, > > If you're interested in Lie groups in a practical sense, then try Peter > Olver's "Applications of Lie Groups to Differential Equations", Springer GTM > #107. Or for a gentler, more concise introduction see Stephani, Differential Equations: their Solution Using Symmetries, (Malcom MacCallum, ed.), Cambridge U Press, 1989. > And other structural aspects of Lie groups lead to the well-known > conservation laws of physics. See Stephani's discussion of Kepler's laws, for instance! > So at least read the intro to Olver's book. Stephani has a similar book > with the word "Symmetry" in its title. See above :-) Chris Hillman ============================================================================== From: Christian Ohn Subject: Re: Lie Group Theory Date: Sat, 22 May 1999 12:16:14 +0200 Newsgroups: sci.math Jason E Tedor wrote: > > Looking for references to "good" introductory texts in Lie Group Theory at > the advanced undergraduate or graduate level. Any suggestions? R. Carter, G. Segal, I. Macdonald, "Lectures on Lie groups and Lie algebras, LMS Student Texts 32, C.U.P. The Lie groups lectures (88 pages) are by G. Segal. They don't come with complete proofs, but they give a very good insight into the theory. -- Christian Ohn email: christian.ohn@univ-reims.cat.fr.dog (remove the two animals) www.univ-reims.dog.fr.cat/Labos/SciencesExa/Mathematiques/ohn/ (ditto) ============================================================================== From: Chris Hillman Subject: Re: Lie Group Theory Date: Sat, 22 May 1999 12:46:55 -0700 Newsgroups: sci.math On 20 May 1999, Jason E Tedor wrote: > Looking for references to "good" introductory texts in Lie Group Theory at > the advanced undergraduate or graduate level. Any suggestions? Lectures on Lie Groups and Lie Algebras, Carter, Segal, and MacDonald, London Math Society Student Texts 32 University of Cambridge Press, 1995 Beware of one slip on p. 55 where g A g^(-1) should be g A g*, where * is the Hermitian transpose of g. (The author was thinking of the action by the subgroup SO(3), where g^(-1) = g^* and we have the spinorial representation of rotations by unit norm quaternions.) Apart from this one mistake I happen to have noticed, I think this is a very nice book. See also the old A.M. Monthly article by Roger Howe for another nice undergraduate level exposition. Chris Hillman