Re: [Phys-l] Simulating a disturbance of a stable planetary system.

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

> On Jan 1, 2008, at 8:19 PM, Ludwik Kowalski wrote:
> > 1) What I would like to learn, from messages on this thread, is
> > how to distinguish a stable system from an unstable system.
> > Don't we need a definition of stability (for the kind of systems
> > being discussed)? If so then what is the acceptable definition?
> >
> > 2) This question is practical, not scholastic. The simulation
> > software I am using , I.P.,  seems to be highly reliable (consistent
> > with underlying physics). How can such software be used to
> > account for unavoidable perturbations? The I.P. does not account
> > for them. Please help me with this issue.
>
> John Denker responded:
>
> "Highly reliable"?  That statement would be more
> informative if it were more specific and more quantitative:
>  -- What tests have been run?
>  -- Quantitatively, how good was the agreement with
>   analytical results?
>  -- Are these tests designed to be incisive?  What classes
>   of bugs are they likely to detect?  Are they appropriate
>   to the numerical methods IP is actually using?
>  -- What are the /limits/ of validity?
>  -- How do we know that IP did not simply incorporate the
>   analytic solution for simple cases?  (That's what I would
>   have done.)  Doesn't that mean  that the results for non-
>   simple cases will be incomparably less accurate?
>
> > How can such software be used to account for unavoidable
> >  perturbations? The I.P. does not account for them.
>
> Sure it can account for them.
>
>  *) For starters, you can run an ensemble of simulations, with
>   slightly different initial conditions.
>  *) Also you can perturb a two-body problem by adding a relatively
>   small third body.
>  *) Et cetera.

1) Last night I responded to the first set of questions.

2) I like the second suggestion above. I may even be tempted to write a 
simulation code in which random perturbations (changes in the state of 
motion) take place, for example, after each step, or after each 
sequence on n steps. But before trying this one must have a definition 
of stability. One must know the criterion for distinguishing a stable 
moving system from an unstable system of similar kind. That is why I 
keep asking for an acceptable definition.

3) The only definition posted so far was my own. Let me describe it 
again, using different words. A moving system, such as two or three 
stars, can be named stable if small perturbations do not destroy its 
initially imposed periodicity. A system in circular motion, for 
example, turning into a system in elliptic motions would be named 
stable. Another system, also initially set up to be cyclic, would be 
named unstable if small perturbations rapidly destroyed its 
periodicity. A system in circular motion, for example, rapidly turning 
into chaotic motion (in less that one perturbation-free period) would 
be named unstable.

 3) What is wrong with this operational definition? Please suggest a 
better criterion for distinguishing stable periodic motion from the 
unstable periodic motion.
_______________________________________________________
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/