From: parendt@nmt.edu (Paul Arendt)
Newsgroups: sci.math
Subject: Lie Groups in ODEs
Date: 26 Sep 1997 20:22:25 GMT

(attributions from sci.physics.research articles; subject was
"Relativity and Intrinsic Properties.")

>>As another aside, it's amusing to note that Sophus Lie himself (of
>>Lie group fame) became interested in Lie groups (he didn't call them
>>that) as a tool to solve (ordinary) differential equations !!!  Many
>>of the unrelated-looking techniques for doing so, like solving exact
>>or homogeneous equations, or taking Fourier or other transforms,
>>all all unified by this: they're really the same technique.  They only
>>look different because the underlying differential equation possesses
>>a different Lie group as a symmetry.

Oz wrote:
>If you can keep it mindblowingly simple could you expand on this a bit?

In a nutshell, if an ODE has a one-parameter Lie group as a symmetry,
it is overspecified in a sense, and can be simplified by taking
advantage of this (reducing the number of variables by one).

The first step is to visualize the space where your ODE lives: make axes
out of all your variables (independent and dependent), and add on axes
which signify all relevant derivatives of dependent variables.  Your ODE
is then some hypersurface (smaller-dimensional manifold) in this space.
I think an example would do nicely here: let's say we have the simple ODE
   y' = y
where y' = dy/dx.  Draw a 3-D space where the axes are {x,y,y'}: the
above ODE is a 2-D surface in this space: a plane which cuts a 45
degree line in a constant-x cross-section of the {y,y'} plane.  OK?

(You've probably solved this in your head already: y = C exp(x), where
C is any number.  But we're going to see, from symmetry reasons, why
this is so easy to solve.)

Now, this ain't no ordinary surface, since the y' axis is special.  Imagine
looking down the y' axis at our surface, and drawing contours of constant
y' upon it (here, they are also lines parallel to the x-axis: lines of
constant y).  There are little tangent vectors induced on this
surface, which represent the direction solutions are going to "travel."
What I mean is, the vector at each point on the surface represents
the slope of the line which is a solution through that point (yes, it's
unique).  However, this is a projection artifact, as we are still looking
down the y' axis: the vectors are really tangent to the surface (as opposed
to being parallel to the y'=constant plane).  Along each of the contours we've
drawn, we will see vectors all pointing the same way: that's what we mean
by a contour of constant y'.

So an ODE is really a "surface" in some higher-dimensional space (which
includes derivatives as axes), with arrows all over it, and always tangent
to it.  In the example above, the surface always looks the same if we move
along the x-axis by any amount.  This is the geometrical picture of the
" y' = y " equation's invariance under the transformation
 x -> x + a
for all real numbers "a".  (This symmetry is the simplest Lie group there
is: the real numbers under addition.)

So we expect that moving in the x-direction by any amount takes us from one
solution onto another solution.  And that's exactly what happens: redefine C
in the solution above, and rewrite the solutions as the three families
                   y = + exp(x + a)
                   y = 0
                   y = - exp(x + a)
where "a" is any real number.

There is a _lot_ to be said about how the solutions split into these three
families, why it was so easy to solve (the symmetry direction was already
a coordinate), and how to use the representations of the group to look
for solutions, but I hope this shows the basic idea.

- Paul