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

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

On Jan 2, 2008, at 11:02 AM, Jeffrey Schnick wrote:

> Ludwik,
> Consider a ball suspended at rest from the ceiling by means of an
> ideal spring.  If I pull the ball down slightly and release it from 
> rest,
> the spring will exert a force on the ball which will be directed toward
> the equilibrium position.  As such, the position of the ball will never
> differ from its equilibrium position by much and its velocity will 
> never
> differ much from its equilibrium velocity of zero.  We call the 
> original
> state of the ball a state of stable equilibrium despite the fact that
> the motion of the ball is forever changed as a result of the
> perturbation.  Stability does not mean that the original state of 
> motion
> is automatically restored.  In the case of a ball on a spring, if the
> oscillations of the ball are small enough, a non-discerning observer
> might report that the ball is still at rest at its original position
> despite the perturbation.
>
> Now consider a low-mass object in circular orbit about a far more
> massive object.  A small perturbation to this system will result in a
> new orbit for the low-mass object but after the perturbation a
> non-discerning observer would probably report that it looks like the
> low-mass object is still in circular orbit about more massive object at
> the same radius as before.  The speed is never much different from
> the original speed and the separation of the objects is never much
> different from the original separation.  It seems reasonable to call
> the original orbit a stable one.
>
> Finally consider three equal-mass objects in which two of the objects
> are in circular motion about the third.  A small perturbation to this
> system results in a state of motion that is so different from the
> original state of motion that even a non-discerning observer would
> report that there has been a drastic change to the state of motion.  It
> seems reasonable to consider the original state of motion unstable.

Let me notice that a ball suspended from the ceiling, initially at 
rest, will not be at rest again, unless dissipative forces (such as air 
resistance) are also present. Any perturbation is a change of the state 
of motion, by definition. The period would not change, after the 
disturbance, but the amplitude (and total mechanical energy) would. The 
disturbance would not change periodicity of such vertical motion. That 
is why I would say that the motion is stable.

_______________________________________________________
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/