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

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

On Dec 31, 2007, at 1:04 PM, John Mallinckrodt wrote:

> Because every instant of time is exactly the same as the initial
> instant of time in terms of the dynamics of this system.  There is no
> need to go to any lengths to apply a perturbation "at some later
> time" or "periodically" or "for a while" or anything else.  Simply
> change any ONE of the 12 initial conditions (i.e., the 2-d position
> and velocity components of the three particles) by a miniscule
> amount, push the start button, and watch the ensuing chaos!

1) That is true, except that the chaos is perfectly reproducible from 
run to run, unless one of the initial conditions was changed before 
each run. There is nothing unexpected in this.

2) What I would like to do, but do not know how, is the following:

a) Watch the undisturbed initial trajectory for a while to show that 
cycles are identical.
b) Disturb the system, during the same run, and observe the consequence.

The expectation is that for a two-body system (planet of mass m and the 
star of mass M>>m) the cycles after the disturbance will eventually 
become identical to cycles before the disturbance, for example, in term 
of periods of repetition. (That would by like disturbing a vertical 
harmonic oscillator by changing g. Increase g and T becomes shorter, 
restore the original g and the original T will restored, after some 
time). For a three body system, such as three stars of equal mass, the 
undisturbed trajectory is also cyclic. But effect of the disturbance is 
expected to be dramatically different, as it is clear to me now.

I would like to be able to perform (a) and (b) using exactly the same 
disturbance. I tried this with one kind of disturbance and it did not 
work very well. However, one thing became clear; for the two-body 
system the motion after the disturbance was still cyclic (an elliptical 
orbit instead of the original circular orbit, as expected). For the 
three-body system, on the other hand, the motion after the disturbance 
became dramatically non-cyclic. It did behave as described by JohnD 
yesterday. That is the best  I could do so far. I cannot imagine a 
single-parameter disturbance, for a two-body system, after  which the 
original circular motion is restored.

Why am I repeating myself? To be sure that nuances are either clear or 
nonsensical. If stability is defined as resistance to changes resulting 
from disturbances, then I failed to demonstrate stability of the 
two-body system. What I was able to do, by using the I.P., was to show 
that the cyclic nature of motion of two-body system was not destroyed 
by the disturbance while the cyclic nature was destroyed by exactly the 
same disturbance applied to a three-body system.  That would be OK if 
stability was defined in terms non-chaotic versus chaotic consequences. 
But this is not OK when stability is defined in terms of absence of 
permanent consequences of a disturbance.

What is the acceptable definition of stability for a system composed of 
two or three particles interacting via central forces?