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*From*: Ludwik Kowalski <kowalskil@mail.montclair.edu>*Date*: Mon, 31 Dec 2007 16:16:02 -0500

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?

**References**:**[Phys-l] two questions***From:*"Anthony Lapinski" <Anthony_Lapinski@pds.org>

**Re: [Phys-l] [tap-l] two questions***From:*William Beaty <billb@eskimo.com>

**[Phys-l] Sharing a problem for students***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] Sharing a problem for students***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] Sharing a problem for students***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] Sharing a problem for students***From:*"Bob Sciamanda" <trebor@winbeam.com>

**Re: [Phys-l] Sharing a problem for students***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**[Phys-l] A numerical simulation of orbiting***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] A numerical simulation of orbiting***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] A numerical simulation of orbiting***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] A numerical simulation of orbiting***From:*John Denker <jsd@av8n.com>

**Re: [Phys-l] A numerical simulation of orbiting***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] A numerical simulation of orbiting***From:*John Mallinckrodt <ajm@csupomona.edu>

**Re: [Phys-l] A numerical simulation of orbiting***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] A numerical simulation of orbiting***From:*John Mallinckrodt <ajm@csupomona.edu>

**[Phys-l] Simulating a disturbance of a stable planetary system.***From:*Ludwik Kowalski <kowalskil@mail.montclair.edu>

**Re: [Phys-l] Simulating a disturbance of a stable planetary system.***From:*"LaMontagne, Bob" <RLAMONT@providence.edu>

**Re: [Phys-l] Simulating a disturbance of a stable planetary system.***From:*John Mallinckrodt <ajm@csupomona.edu>

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