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

[Bernard Cleyet <bernardcleyet@redshift.com>]

I think first examining a much simpler system as Eisberg & Lerner do, 
would be more heuristic.

E & L define stability as reacting to perturbations w/ out "spiralling" 
out to infinity or in to the "sun".  I suppose a result may be "chaotic" 
if the orbit never repeats *.  Depending on the meaning of "bound" the 
instability that results in decay is still bound.  (E < zero.)

W/ 1/r  central force, precession is required (How Hooke showed the G 
force was likely 1/r^2 before Newton).  I'll bet it's a space filling orbit.

bc wishes Ludwik would not append his replies, but "prepend" them.

p.s. I suspect there are analytic methods to deal w/ your problem, but 
certainly much beyond my ability.  I can't find the one celestial 
mechanics text I have to perhaps verify.

The person missing from this discussion may have used I.P. similarly -- 
however not at the site I read:

http://www.csupomona.edu/~ajm/ip.html#orbitens 
<http://www.csupomona.edu/%7Eajm/ip.html#orbitens>


Evidently I.P. compatible platforms do not include OS X!

http://www.design-simulation.com/IP/faq.php#MacOSX

Yes, I mentioned R-K only to illustrate the prevalence of sophisticated 
numerical work in recent * texts.

And @ $250/ single use, and not OS X native, forgetaboutit.


* as opposed to mid 20th cent.


Dear Bernard,
1) I hope the issue of stability of the three stars system, or a 
similar system of two positive charges orbiting a negative charge, will 
be resolved without using numerical simulations. But I the issue of 
possible bugs in a general purpose-simulation code, mentioned by JohnD, 
is worth pursuing, if you have time. The only advantage of the 
Runge-Kutta, if I remember correctly, is that a smaller number of 
steps, for example 10000 instead of 100000, is required for the same 
accuracy. That was a big issue when computers were slower, and less 
accessible.

2) I do not think that a bug in Interactive Physics was responsible for 
my difficulties. But I will be glad to change my mind about this.  I 
would do exactly what you do using the I.P. and we would compare the 
results.

Ludwik



Ludwik Kowalski wrote:

> 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/
>
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