On the Bautforum there's actually a thread going on about the stability of the Lagrangian points of the Earth and Moon .
http://www.bautforum.com/questions-answers/74989-orbit-question-3.html Theres a reference to the Neil Cornish paper which calculates the Lagrangian points
http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf With the formulas in this paper one can calculate the position and velocities of the bodies residing in the L1,L2,L3 points : ( referenced to the mass center of Earth/Moon )
"Earth", -4665880.79, -12.455,
"Moon", 379334119.21, 1012.617
"MoonL1", 321375133.17, 857.90
"MoonL2", 443782166.58, 1184.659
"MoonL3", -385944079.29, -1030.262
Using the following data ( as Tony did )
For Earth: 5.9736E+24 kg
For Moon: 7.34764E+22
G: 6.6725985E-11
R =384000000 m
All of the points are known to be unstable. But how "unstable" ?
Tony is correct as he says the L3 point is rather stable . (L3 being the point opposite to the moon ) .
To find out the degree of instability the above data were input in the Picard Integrator and ran for 1 year . Screenshots were made every 1/100 year.
One can see how all the test masses leave their original position after a short time .
L3 keeps its position for a rather long time . L1 seems to become a moon of our moon .
I'll try to post a rotating frame also which is far more expressive ...
( hint : click on the shortcut to open a separate frame )