Assume there are only three celestial bodies in the universe: the Earth, the Moon, and a body with a mass so small that it does not affect the other two appreciably. Even under these circumstances it is not possible to find an exact equation for the orbit of the small body. In fact, the orbit may exhibit chaotic behavior, i.e. even small variations in the inital conditions will be amplified.

There are two exceptions: The French mathematician Lagrange showed in 1772 that if a small body is placed in the orbit of the Moon so the angle Moon-Earth-body is 60 ^degrees, and with the same velocity as the Moon, then it would continue to be in this position relative to the Moon. These two points are called the Lagrange points L4 and L5 and are points of stable equilibrium. A body in the neighborhood of one of these points will continue to stay in that neighborhood indefinitely.

Near  L4 and L5 in the  orbit of the Moon, only loose clouds of small particles exist. Near the corresponding points in the orbit of Jupiter we find more than 200 asteroids, called the Trojan asteroids, which have probably strayed into the Lagrangian points from the main asteroid belt.

In this simulation Body60 is placed in L4 and Body90  90 ^degrees away from the Moon. Both have the same initial velocity as the Moon.

(Also see the simulation "Satellite near L4").

1. Run the simulation and study the orbits of Body60 and Body90.
   
   
2. (Advanced). A difficult mathematical analysis reveals that if the mass of the Moon were greater than about 1/25 of the mass of the Earth, L4 and L5 would have been unstable. Calculate the velocity of the Moon if its mass were 1/10 of the mass of the Earth (see the simulation "Orbit of the Moon"). Also change the velocities of Body60 and Body90 and run the new simulation.