There are some interesting orbits in the 3-body simulation. For example, are there any orbits that are stationary relative to the other objects. Such orbits would be of interest for parking satellites, rockets and supply depots. It turns out there are such orbits, the most obvious one being an orbit between a planet and a moon nearer to the moon but where the speed of the satellite is proportionally smaller than the speed of the moon. As we have seen in the earlier examples, at lower altitudes relative to a planet we need a higher speed to stay in orbit. However, somewhere between the planet and the moon there is a point where the moon's gravity will stop the satellite from falling to the planet. This point is known as the first Lagrange point, denoted as L1 orbit. These stationary orbits are name after Joseph-Louis Lagrange the 18th-19th century mathematician who first explored the mathematics of these orbits. Unfortunately the L1 orbit is unstable, as can be seen in this simulation. An arbitrarily small pertubation in the orbit will cause the rocket to either fall towards the planet or ascend towards the moon. Such an orbit can be sustained for just over one cycle around the planet, but will eventually require thrusters to correct the pertubations to prevent it from deviating away from the planet. You can explore the effect of deviating from the L1 height and speed to see what effect these deviations have on the trajectory of the satellite.
Below the simulation you will see 5 buttons, labelled L1 to L5. Click on these buttons to set the simulation up for each of the 5 Lagrange points between the planet and the moon. Note the difference in stability characteristics in each of these orbits. Suprisingly, the orbits L4 and L5, not between the planet and the moon, are the only two stable Lagrange points. When the satellite is parked near one of these points it will actually orbit the Lagrange point itself. If you watch the readouts of the Satellite slider you will see that the energy of the satellite is conserved. You can see that as the distance from the satellite to the planet decreases, so the speed correspondingly increases, and as the distance from the satellite to the planet increases, the speed correspondingly increases.