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There are Asteroids "trapped" in Jupiters Langrange points 4 and 5 called trojans and greeks. Are there any asteroids in the earths L4 and L5? Have we seen asteroids in Lagrange points of the earth and moon system? Why are only L4 and L5 stable and L1, L2 and L3 require corrections to remain at those positions?

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There is one known earth Trojan - http://en.wikipedia.org/wiki/Earth_trojan

The reason L1, L2 and L3 aren't stable is covered here: https://physics.stackexchange.com/questions/36092/why-are-l4-and-l5-lagrangian-points-stable

The reason the earth has so few L4 and L5 asteroids is cause other planets like Jupiter and Venus tend to destabilize the orbits - same reason why Jupiter has so many.

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