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The Lagrangian points are points of unstable balance (at least gravitationally; L4 and L5 are stable thanks to Coriolis force.), and that means an object not stabilized actively will fall out of them and start accelerating down the steepest gradient of whichever gravity well it toppled into.

But with the gravity wells being as irregular as they are...

enter image description here

and the object gaining momentum, never mind neither of the two bodies remaining at rest, the trajectory becomes less intuitive.

So, assuming a minimal nudge, just a smallest loss of stability, where would the object end up starting at different Lagrangian points? Which nudges out of L-points make the object crash into Earth or the Moon, which will let it escape the 2-body system, and will any set it on a mostly cyclic orbit? (possibly unstable due to tidal forces, but still one that will take some time to decay.)

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  • $\begingroup$ Related: If something “falls off” the L2 or L1 point, where will it go? $\endgroup$ – TildalWave Dec 5 '15 at 0:13
  • $\begingroup$ Hm. The related question deals with L 1 and 2. The dynamics in the other three look different. To the extent the answer is probabilistic it would be the same for all of them, however the most likely outcomes must be different for L 4 and 5, and especially 3. $\endgroup$ – kim holder Dec 5 '15 at 1:46
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    $\begingroup$ Nudging from EML1 can give a wide variety of results: crashing into the moon, crashing into the earth, or expulsion from earth's Hill Sphere. Same goes for EML2. Nudging from EML3, EML4 and EML5 seems to result in longer lived horseshoe orbits that stay at a lunar distance from earth, sometimes appraching and then receding from the moon on the trailing side, sometimes approaching and then receding from the moon's leading side. $\endgroup$ – HopDavid Dec 5 '15 at 3:33
  • $\begingroup$ @HopDavid: How can L1 lead to escape? What would be the trajectory for that? $\endgroup$ – SF. Dec 5 '15 at 9:14
  • $\begingroup$ See my answer here. $\endgroup$ – honeste_vivere Dec 5 '15 at 15:08
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EML1 and EML2

From EML2 and EML1 it is possible to collide with earth or the moon. It is also possible to sail out of earth's Hill Sphere.

Here are a range of pellets from EML2 nudged away from the earth and moon:

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Some sail out of earth's Hill Sphere. Note the orange pellet makes a close approach to earth.

Here are some trajectories from EML1:

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Nudging towards the earth results in an approximately 300,000 x 100,000 km earth orbit. Third or fourth apogee the moon pulls the pellets backward lowering perigee. Over time it is possible for pellets to collide with earth.

Here are pellets from EML1 nudged towards the moon:

enter image description here

Notice at 3rd apolune they sail through EML2 out towards the outer reaches of earth's Hill Sphere.

When nudged moonward they fall into an about 60,000 km x 5,000 km lunar elliptical orbit. While this ellipse's line of apsides remains constant, EML1 and EML2 rotate. At the third apolune the pellets find themselves in the neighborhood of EML2.

A small braking burn at that 3rd apolune could park at EML2. Thus it is possible to go from EML1 to EML2 with little delta V. And vice versa.

EML4, EML5, EML3

I believe something nudged away from a Trojan point will form a tadpole orbit:

enter image description here

Bigger the nudge, the bigger the tadpole.

If the tadpole's tail extends past EML3, you get a horse shoe:

enter image description here

If the earth, moon and pellet were an ideal 3 body system, these tadpole and horseshoe orbits could be long lived.

But the sun is a major influence. The sun's influence can wreck what would otherwise be fairly stable orbits.

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