On 1/11/13, rcs@xmission.com <rcs@xmission.com> wrote:
Warren's proof would seem to prove that the equilateral triangle configuration is unstable. My impression is that it's at least neutrally stable (for some mass ratios). I haven't done the details, so I'll leave this as a question, rather than an assertion of fact. [One possible out: the rotation might be non-uniform?]
Sometimes communication is unclear, and CAPITAL LETTERS don't really help. By reading Mr Smith's description, and knowing what I do about Lagrangian solutions L_4 and L_5 [1], I believe that Smith does not consider them "STABLE" (in capital letters :-) because of the "kidney bean shaped" paths that a perturbation must necessarily produce in order for it to be a kinetic-energy-conserving solution. Of course, the L_4 and L_5 solutions are stable (without capital letters) because there is an attractor. But that's not good enough for Smith's criteria, which I think Meeker was confirming. - Robert Munafo [1] Pretty much limited to http://en.wikipedia.org/wiki/Lagrangian_point#Stability and the proof-of-existence demonstrated by Trojan asteroids
On 1/10/2013 3:53 PM, Warren Smith wrote:
Suppose you have N planets moving under Newton's laws of gravity& motion (treat as point masses). Further, suppose that in an appropriate -- rotating about the center of mass -- reference frame, all the planets are stationary. Finally, let the configuration be STABLE against small perturbations.
MY CLAIM: If N>2 then no such configuration exists.
[...]
Right. Perturbing a mass at an L4 or L5 causes it to make a small orbit around the Lagrange point. Of course this also causes a small perturbation in the motion of the two larger bodies.
Brent Meeker
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com