Re: [Phys-l] a nonlinear ODE

[John Denker <jsd@av8n.com>]

On 12/28/2008 06:30 PM, Carl Mungan wrote:
> > Consider the following differential equation for motion in a single
> > variable x:
> >
> > x*a=k*t                     [1]


Question:  What physics leads to that equation?  I've
tried and failed to guess where that comes from.  Or
is it just recreational mathematics?


On 12/29/2008 01:17 AM, I wrote:

> There is one rule to live by:  Start by drawing a picture.
> I plotted some solutions to equation [1] for k=1 and
> various initial conditions:
>   http://www.av8n.com/physics/img48/xddx.png

And continuing that thought, here's an example of why you
want to draw the picture.  Just by staring at the picture
it seemed to me that all the curves might have a similar 
asymptotic behavior.

You guys should know me well enough by now to guess what
comes next.  Scaling!

Equation [1] exhibits some simple and informative scaling
behavior.

Substitute 
      y := x / s^3
      τ := t / s^2
then
      y (d/dτ) (d/dτ) y = k τ

which has the same form as equation [1], with the same k.
The only difference is that we have "zoomed out" by scaling
the variables by factors of s^2 and s^3.

This is interesting because
       y(0) = x(0)/s^3
      (d/dτ)y(0) = (d/dt)(0)/s
which means that the y-equation has less y-intercept and
less initial slope.  In the limit as we zoom out very very 
far, we are left with the solution that has no intercept 
and no initial slope, i.e. the power law that Carl found, 
shown in green in my picture
  http://www.av8n.com/physics/img48/xddx.png

The scaling argument tells us that *all* solutions of
equation [1] go like t^1.5 asymptotically.  In fact
we even know the factor out front:  they all look 
like sqrt(4k/3) t^1.5 if we zoom out enough.


============================

And now for some pedagogical and philosophical remarks:
Scaling has been an important part of physics since Day One 
of modern science (1638) ... and is still very important:
  http://www.google.com/search?q=scaling+site:nobelprize.org

Scaling laws have many uses.  Scaling laws are easy to learn 
and easy to teach.  Indeed they are more useful and more age-
appropriate than a lot of the stuff that is typically taught.
  http://www.av8n.com/physics/scaling.htm