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Re: [Phys-l] Phys-l Digest, Vol 36, Issue 1

On Jan 1, 2008, at 5:23 PM, Ludwik Kowalski wrote:

On Jan 1, 2008, at 2:10 PM, Alfredo Louro wrote:

If you change the initial conditions, you must get a different
stable orbit, I think. There is no reason why it should decay
back to the original initial conditions before you perturbed
it. It seems to me you are solving two different problems.

I agree. That is why I now think that defining stability in terms of
"returning to the original state of motion" (state of motion before a
small perturbation was applied) is not reasonable. In view of my
experience with I.P. simulations, I now think that stability, in
situations being discussed, should be defined in terms of the loss of
periodicity. For a two-body system periodicity is not lost (a singular
circular orbit is replaced by two coupled elliptical orbits), but in
the case of the three-body system, periodicity is lost and motion
becomes chaotic (after the same kind of perturbation). All this can be
simulated in I.P.
P.S.
The first thing I did, in this year 2008, was to simulate a pencil
standing vertically on its sharp point. We know that, in reality, this
kind of equilibrium is unstable. But the pencil on my screen remained
vertical. Why was it so? Because the I.P. code ignores minute
disturbances which are always present. The pencil started falling when
a tiny additional weight was places on a side of its flat upper
surface. I love I.P.

P.P.S.
A single small disturbance of a two-body system (two identical stars on an initially-circular orbit) will produce two slightly elliptical orbits. But suppose small perturbations are random and tend compensate each other's effect. What is the cumulative effect of many disturbances? Will the circular orbit be restored? Yes, this is possible. But the probability would be very small. I am thinking in terms random walk, where the probability of returning to the point of origin is very small.
_______________________________________________________
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/