Re: [Phys-l] a nonlinear ODE

[John Denker <jsd@av8n.com>]

On 12/29/2008 10:02 AM, Carl Mungan wrote:

> The scaling argument is cute, but how did you come to choose to 
> divide x by s^3 and t by s^2?

A reasonable question.  I posted the result without
explaining where it came from.  FWIW for me this stuff
is like riding a bike:  It's easier to do it than to
explain it ... but I can explain it if need be.

So here's the procedure:  scale x by its own factor p,
and scale t by its own factor q:
    x = y/p
    t = τ/q

Then plug into equation [1].  Then collect factors of
p and q and see what you have to do so that the new
equation has the _same form_ as the old equation.
(If you take logarithms, the task begins to look a lot
like balancing a chemical reaction equation.)

Actually you could have known before we started that
p would go like the 1.5 power of q, because of the
solution x ∝ t^1.5 that we already had.

I rewrote 1.5 as 3/2 and wrote p as s^3 and q as s^2.


> You appear to have 
> rigged up yours to give less intercept and slope, is that the 
> criterion you used to choose them?

The _criterion_ is that the equation shall be scale-invariate 
with respect to changes in the scale-factor s.

OTOH I was very unsurprised to find that large s gave me a
small y-intercept and initial slope.  I was expecting this
based on playing with the spreadsheet, looking at zoomed-out
versions of the picture I posted.

 
> In any case, you have convinced me that I have found the asymptotic 
> solution. Let's define A(t)=sqrt(4k/3)*t^1.5 (A for asymptotic). Can 
> that help us find the general solution to my original equation 
> [f(t)=k*t]? I'm thinking of writing something like x(t)=[T(t)+1]*A(t) 
> where T(t) is a transient that decays to zero as t->infinity. This 
> approach just leads me to a mess when I sub it into my ODE. But maybe 
> there's another approach someone can suggest?

I would not recommend writing the general solution as (something)*A(t).
The problem is that A(t) goes through (0,0) which makes it very hard
to describe what the general solution is doing in the small-t regime.
And that's where all the action is;  we already understand the large-t
regime (via the scaling argument).

There's a lot of other things you could try, e.g.
  z(t) + t^1.5
or
  (z(t) + t)^1.5
or
  z( t^1.5 )

Those at least have a chance of nontrivial small-t behavior along
with the correct large-t behavior.  I don't actually expect those
to help much, i.e. I expect the equation for z(t) to be a mess,
but it doesn't cost much to try.

=========

I ask again:  How badly do you need a closed-form solution to this
equation?  In this business, it pays to take a step back and ask
what it means to "solve" a problem.  The conventional and sensible
advice is that your priority should be to _understand_ the what
x(t) is going to do.  Understanding does not mean that it is
necessary or even possible to write down the solution in closed
form in terms of known functions.

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

On 12/29/2008 10:42 AM, John Mallinckrodt wrote:

> > 2. The "stiffness" of the DE near x = 0 suggests pathological  
> > behavior.  Clearly, any launch toward the origin will be repelled,  
> > but the question arises: Is it possible to launch with a high enough  
> > initial velocity toward the origin to "overcome" the repulsion and  
> > pass to the other side?

This is a fine example of the sort of "understanding-oriented"
questions that can be formulated and answered without reference
to any closed-form solution.

> >   It is easy to see that the integral from any given value of x  
> > to the origin diverges logarithmically, suggesting pretty  
> > conclusively that no finite initial velocity can overcome the barrier  
> > at the origin.  For large enough velocities, the trajectory  
> > essentially bounces off the origin.

There's another way of stating this result: Energy!
Think about the KE and PE.  The PE barrier is infinite at x=0.

This approach is fairly general.  That is:
A lot of differential equation systems, even nasty nonlinear
time-dependent ones, have energy-like _integrals_ that can
be exploited in this way.  These aren't automatic or free;
that is, often you have to do a bunch of work to _construct_
the thing you want.

One category of such things goes by the name of Lyapunov 
functions.
  http://en.wikipedia.org/wiki/Lyapunov_function

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

There is a beautiful passage in Feynman where he starts with a
certain second-order differential equation and shows that there
are two solutions, denoted s() and c().  He derives from scratch
the properties of s() and c(), including the fact that they are
periodic and that s^2(t) + c^2(t) = 1 for all t.  There is are
also rapidly-convergent approximations for s() and c().  The
point is that you can know everything you need to know about
s() and c() even if you cannot "solve" the differential equation 
in terms of known functions. i.e. even if you have never heard 
of sin() and cos().

Feynman was fanatical about not relying on the names of things.
He insisted that it was infinitely more important to know what
something did, and infinitely less important to know what its
name was.  He avoided standard names even when things had standard 
names, which IMHO is going too far, because if you know the 
name you can look it up to find out what it does.  But still
he had a mostly-valid point, especially from his own perspective:
he was so far out on the leading edge that most of the stuff
he encountered didn't have names.  So he *needed* to figure
out how it behaved, from scratch.

I don't know whether there exists a closed-form expression
for the general solution to equation [1].  Maybe it exists,
maybe it doesn't.  Maybe it's worth the trouble, maybe not.
Maybe it will turn up in the medium-term or long-term.

In any case, in the short term, it appears that we don't 
yet have a closed-form solution.  Therefore in the short 
term (and maybe longer), this is a good opportunity to put 
on your wizard hat and practice _understanding_ equations 
that you don't have a closed-form solution to.