What are Lagrange points in gravitation fields? 
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*What are Lagrange points in gravitation fields? 

*Additionally, what are their properties? For example, if a satellite or asteroid rested at a Lagrange-point.
 A: A Lagrange point is a position relative to a system of two-body gravitational objects at which a third negligible small object, if its velocity is correct too, would not move relative to the two other objects. The two gravitational objects will orbit around the center of mass of both objects together (also called the barycenter). I am not sure, but I also believe that this orbit should be (near) circular. This means that the third object would have to orbit orbit this center of mass as well. So the gravitational forces, exerted by the two gravitational objects on the third object, should add up to the required centripetal force to keep it on a uniform circular motion, with the same angular velocity as the two gravitational objects around each other.
However these points are only a mathematical equilibrium. If your position or velocity would be off just  a little bit you would eventually move away from most point. Namely points $L_1$, $L_2$ and $L_3$ unstable because they are saddle points. The remaining two point, $L_4$ and $L_5$, are stable. You would not move away from it if your position of velocity is off, but keep orbiting around it.
So in case of the two body system containing the Earth and the Moon a small asteriod or man-made satellite would have a negligible small mass. So if they where at the position of a Lagrange point (including the correct velocity), they would remain there for a while in case for the first three. And they would remain there indefinitely in case of the last two if they would not experience to much perturbation, for example due to solar radiation or the gravitational pull of the massive gas giant Jupiter. But once close to such a point a satellite could occasionally perform small correction burns to keep itself near it. Such as the Gaia space observatory oscillates around the Sun–Earth $L_2$ Lagrangian point.
