How to use Lie Groups in differential equation

David Billinghurst wrote:
> There is some code in share/contrib/diffequations that solves 
> first order ODEs using Lie groups.  It works - most of the time
> - but there are still some bugs to shake out.
> Check out the references and the testsuites.

I am repeatedly pleasantly surprised to find
out how much Maxima does include.

Below are the references from ode1_lie.mac (in
the directory David recommended):

  [1] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order 
      ODE Patterns, Computer Physics Communications 113 (1998), p 239. 
      (http://lie.uwaterloo.ca/papers/ode_vii.pdf)
  
  [2] E. S. Cheb-Terrab, T. Koloknikov,  First Order ODEs, 
      Symmetries and Linear Transformations, European Journal of 
      Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). 
      (http://lie.uwaterloo.ca/papers/ode_iv.pdf)

  [3] G W Bluman, S C Anco, Symmetry and Integration Methods for
      Differential Equations, Springer, (2002)

I found a bit more by adapting my Google search to 
(all on one line):

 [ Symmetry | lie | Integration "differential equations" | ode | pde | "partial differential" ]

http://www.scg.uwaterloo.ca/~ecterrab/papers/

Program LIE for Lie analysis of differential equations on IBM type PCs
http://archives.math.utk.edu/software/msdos/adv.diff.equations/lie/.html
Reference books and articles from this program:
"Applications of Lie Groups to Differential Equations" by P. J. Olver
 ( and the worked examples in Chapters 2 and 3 )
"Symmetries and Differential Equations" by G. W. Blumen and S. Kumei
"Symmetries of Differential Equations: from Sophus Lie to Computer Algebra" 
  by F. Schwarz, SIAM Review, vol 30, Sept 1988, pp450-481. 
Many of the examples in the DAT files are from these sources. Source code is included. 

How do Lie groups enter the analysis of a differential equation? 
http://www.math.niu.edu/~rusin/known-math/97/lie.ode 
Applications of Lie theory for solving differential equations 
http://www.math.niu.edu/~rusin/known-math/99/lie_diff
How do Lie groups and algebras fit in among the branches of mathematics? 
http://www.math.niu.edu/~rusin/known-math/99/lie_algebras

Lie Groups of Partial Differential Equations and Their 
Application to The Multidimensional Screening Problems 
http://eprints.unimelb.edu.au/archive/00000584/01/895.pdf

Another search:
 [ Symmetry | lie  pdf "differential equations" | ode | pde | "partial differential" -review ]

ALGORITHMS FOR DIFFERENTIAL INVARIANTS
OF SYMMETRY GROUPS OF DIFFERENTIAL EQUATIONS
http://oregonstate.edu/~pohjanpp/kpdi.pdf

http://www.math.missouri.edu/~kcshin/res.pdf

Modifying the search again:
 [  "lie groups" | "lie algebra"  pdf "differential equations" | ode | pde | "partial differential" -review ]

Sophus Lie and the Role of Lie Groups in Mathematics By Sigurdur ...
http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-755Fall-2004/B207F5F1-43A8-48A2-B7CC-C4D8C2794041/0/helga_sophusl1_1.pdf

Exponential Map and Differential Equations on Real Lie Groups
http://www.emis.ams.org/journals/JLT/vol.13_no.1/moskola2e.pdf

Hidden Symmetries, First Integrals and Reduction of 
Order of Nonlinear Ordinary Differential Equations
http://www.sm.luth.se/~norbert/home_journal/9s2-1.pdf


Reduction of Order for Systems of Ordinary
Differential Equations  (Short, doesn't seem to advanced)
http://www.sm.luth.se/~norbert/home_journal/electronic/111art2.pdf


14 chapters and two appendices of the draft of a shortened version 
of the text "Lie Groups, Lie Algebras, and Some of their Applications"
by Robert Gilmore 
Geometry and Lie Groups
http://www.physics.drexel.edu/~bob/LieGroups.html


The Life and Times of Emmy Noether
Contributions of Emmy Noether to Particle Physics???
http://www.physics.ucla.edu/~cwp/articles/9411110.pdf

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Herb Martin, MCSE, MVP
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