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



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.