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



On Dec 30, 2007, at 9:40 AM, John Denker wrote:

On 12/30/2007 08:14 AM, Ludwik Kowalski wrote:
.... Keep
in mind that I am asking this questtion in the context of a specific
problem. Three identical stars are initially at rest on the diameter of
a circle (2*R) .....

This problem is neither stable nor unstable.
This problem is neither chaotic nor non-chaotic.
This problem merely "is". The solution is unique. It just "is".
Stability has to do with /perturbations/ but this problem
as stated doesn't allow any perturbations.
Chaos has to do with sensitivity to initial conditions
(plural) but this problem as stated has only one initial
condition.

3) What is wrong with saying that an orbiting system, described above,
is stable when it is energetically bound?

The problem is so narrowly specified that stability
questions cannot be asked, let alone answered.

4) Why am I insisting on limiting the discussion to a specific
three-body system? Because the circular orbit system is mathematically
simple.

It's too simple.

The fact that a moving system can possibly be unstable (or chaotic)
does not mean that it actually is unstable or chaotic.

True.

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

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 ?

Yes I know that:

a) the stars would accelerated toward each other (along the diameter, and form a three time larger star in the center, if initial speeds were zero.

b) the stars would move further and further away from each other (and never come closer) if initial speeds were much larger (large enough to make E positive).

c) the outer stars would turn around the central star along elliptical trajectories if v were larger than zero but smaller than what is needed for a circular orbit. The central star would be at the focus of each ellipse.

ANOTHER QUESTION:

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?

How to
distinguish a set of initial conditions that produces a stable moving
system from a set of conditions that produces an unstable system?

In general, map the parameter space.

For example:
http://en.wikipedia.org/wiki/Julia_set

This would be too abstract for me and for my students. I suspect that many would feel the same way. I was probably a typical teacher and my students were probably typical students.

_______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/