How to use Lie Groups in differential equation

> 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. 

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.

> My mathematics training is 1 1/2 years university level as part of a 
> bachelor of science, and then self-trained. I'm up to date on 
> differentiation, calculus, vectors, matrices, complex numbers and 
> quaternions, but I've had very little training in 
> mathematical symbolism.

With a similar (or slightly weaker) background, 
I became interested in these topic through reading
"The Road To Reality" by Penrose.

> So, does anyone have any links to free resources that covers 
> this topic 
> at a 2nd/3rd year university level? Preferably with some 
> worked examples? :)
> 

Lie Symmetry Analysis of Differential Equations
http://www.mat.itu.edu.tr/can/BOOKS/symmetries/e-book.htm
Many authors -- Separate PDF files

Computer Algebra Solving of Second Order ODEs
Using Symmetry Methods
http://www.scg.uwaterloo.ca/~ecterrab/papers/ode_ii.pdf


What IS a Lie Group? 
<http://www.valdostamuseum.org/hamsmith/Lie.html>; 

What is a Lie algebra, really
A geometric view
<http://www.math.siu.edu/kocik/lie/lie-map.htm>; 

Learn the content of the Lie mandala
<http://www.math.siu.edu/kocik/lie/learn-a.htm>; 


Geometry and Group Theory
Christopher Pope (TAMU)
181 pages -- Manifolds, GR, Lie Groups & Algebras
http://faculty.physics.tamu.edu/pope/geom-group.pdf

An Elementary Introduction to Groups and Representations
Brian C. Hall -- Lie Group and Representation Theory
http://arxiv.org/abs/math-ph/0005032
128 pages

Introduction to Groups, Invariants and Particles
Frank W. K. Firk, Professor Emeritus of Physics, Yale University
162 pages -- mostly Lie groups
http://www.physicsforfree.com/intro.html
http://www.vegetarianusa.com/physics/introgroups.pdf

Groups and Symmetry
Andrew Baker
http://www.maths.gla.ac.uk/~ajb
68 pages -- not specifically Lie groups

Group Theory: Lie's, Tracks, and Exceptional Groups
Predrag Cvitanovi
http://aux.planetmath.org/files/books/60/GroupTheory.pdf
pages 283 -- too advance so far, but I really want to read this

Gravitation, Gauge Theories and Differential Geometry
T. Eguchi, P.B. Gilkey and A.J. Hanson
178 pages -- recommend by Geometry and Group Theory (above)
http://www.maths.ed.ac.uk/EMPG/Activities/GT/EGH.pdf

Topics in Differential Geometry -- (359 pages)
Peter W. Michor
http://www.mat.univie.ac.at/~michor/
http://www.mat.univie.ac.at/~michor/dgbook.pdf
http://www.mat.univie.ac.at/~michor/dgbook.ps


At the following link you will find a large list
of both free online and published books for math:

The Return of the Mathematics Corner
George E. Hrabovsky
<http://www.sas.org/tcs/weeklyIssues/2004-10-15/mathcorner/index.html>; 
Partial Differential Equations: The online free books for this are: Birnir's
"Elementary Partial Differential Equations and Applications," Showalter's
"Hilbert Space Methods for Partial Differential Equations," and Herod's "An
Introduction to Partial Differential Equations."

Differential Geometry Reconstructed: a Unified Systematic Framework
Alan U. Kennington
http://www.topology.org/tex/conc/dgstats.php
(I am currently reading this.)

Please send references to any additional notes 
or resources you locate.

--
Herb Martin, MCSE, MVP
HerbM at LearnQuick.Com http://LearnQuick.Com
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