Re: [Phys-l] A numerical simulation of orbiting

[John Denker <jsd@av8n.com>]

On 12/30/2007 11:48 AM, Ludwik Kowalski wrote:

> OK, let me ask two different questions. Suppose I am able to perform a 
> described experiment, somewhere far away from any galaxy.

This is a significantly different scenario than what went 
before in this thread.  That's OK by me;  we just need to 
be clear about which scenario is which.  Things that are
true in one scenario are not true in another.

I emphasize that this is now a "physical" physics situation 
(in contrast to the previous "mathematical" physics situation)
so this system will be subject to real-world perturbations.

> 1) Is it true that the closed system of my three stars will start 
> rotating as described ?
> 2) Is it true that the described circular motion will continue, cycle 
> after cycle ?

Too vague.

1) First of all, it would be helpful to consider an /ensemble/
of such systems.  An ensemble of two (a duet?) suffices.
Take one of them to be perturbed, and the other to be
unperturbed (or much less perturbed) and do the comparison.

2) You need to ask what happens after N cycles, for various 
values of N.  In a chaotic system, the separation between
one trajectory and another is an exponential function of N.

So the two members of the ensemble would start out similar
and would stay similar for a while, possibly hundreds of
cycles or even more ... but then would diverge, losing any 
semblance of similarity.


> 3) Would it be desirable to present this kind of gedankening to 
> students of introductory physics course, or at least to a group of 
> selected students, to promote critical thinking and learning?

Yes.  The mathematics of iterated nonlinear maps 
  http://complex.upf.es/~josep/Chaos.html#The_logistic_map
is well within the reach of high-school students.  Take a 
look at the bibliography
  http://www.visual-chaos.org/refs/texts.html
and search for "high school".

You can use the class as a parallel computer:  When
studying the equation
   x(t+1) = r x(t) [1 - x(t)]
  
you can assign each member of the class two values of r 
(one chaotic and one not) and assign suitable random 
initial x-values.  Then have them grind out 100
iterations. Collect all the data and plot it, resulting 
in a nice diagram such as
  http://complex.upf.es/~josep/bifurcation.jpg

If students are able to use spreadsheets, this is super
easy, but even with just a hand calculator it is feasible.

Better yet, assign each students three r-values
 a is non-chaotic
 b is chaotic
 c is close to b, but differs by 1e-10.  This provides a 
  well-nigh unforgettable lesson as to what we mean by 
  extreme sensitivity to initial conditions.

If this seems too simple, don't worry.  One of the
triumphs of chaos theory is to show that many aspects
of chaos are /universal/ i.e. seen in /any/ chaotic
system.  So studying an ultra-simple chaotic system
is time well spent.