How to use Lie Groups in differential equation

Hello,

> > Hi all. I remember somebody saying in this news group that Lie Groups 
> > (symmetry analysis) was a general purpose approach to solving 
> > differential equations. I've been googling the web for a week now in 
> > order to understand how this works, and now have a bit of an 
> > understanding of what an algebra/group is (and specifically what Lie 
> > algebras and groups are).
> 
> Most of my links (below) are NOT specific to DiffEq,
> with the first couple as a major exception.  I seriously 
> doubt you will find much at an undergraduate level.
> 
> I will be delighted to see other answers to your
> request as this is a topic I am currently studying
> also. 

I was involved in symmetry analysis sometimes in my studies. The book which is
most elementary and contains less `mathematical symbolism' is the book by
Bluman and Kumei: Symmetries and Differential Equations (Applied Mathematical
Sciences, Vol 81) Hardcover: 412 pages
Publisher: Springer; Corr. 2nd print edition (August 1989)
Language: English ISBN: 0387969969

You might try to find it used at www.bookfinder.com
There are many other books, some very famous, but this is the one that I think
should be better for you.

The old Macsyma had a package for computing symmetries and much more: its name
was symmgrp.max, here you will find its page:
http://www.mines.edu/fs_home/whereman/liesym.html
It would be *very* nice to be able to run it in the actual version of
Maxima. Unfortunately I'm not a programmer and I'm not able to realize how much
it would need to port it. I don't now if the authors would allow this
either.

> 
> When searching, try adding "PDE | "Partial differential"
> to you criteria as most of this will likely be related
> to those types.
> 
> > However, each time I try to understand how Lie algebra applies to 
> > differential equations, I slam into a wall of mathematical symbolism.
> 
> The best CONCEPTUAL explanation of Lie Groups and
> Algebras was not directed at solving ODE/PDE problems
> but rather at explaining how the Lie Groups relate 
> to Quantum Field Theory and specifically Gauge Theory:
> 
> 	"Deep Down Things: The Breathtaking Beauty of Particle"
> 	Bruce A. Schumm 
> 
> Schumm has 3-4 chapters in this book related to understanding
> and using Lie Groups in QFT/Gauge theory, although this is
> not what you are requesting it is the best popularization
> of the subject that I have found.

I do not know the above texts, but I think that the way in which Lie groups are
used can be very different. In particular, the particle physics applications
are far from the symmetry analysis of differential equations.

Bye, raf.

-- 
Raffaele Vitolo, Dipartimento di Matematica 'E. De Giorgi'
Universita' di Lecce, via per Arnesano 73100 Lecce ITALY
tel.: +39 0832 297425 (office) fax.: +39 0832 297594
home page: http://poincare.unile.it/vitolo