Gravity lab
Pick any pair of bodies and watch gravity itself. The view rides along with the pair, so both sit still while the whole frame spins; gravity plus that spin makes a landscape with five ledges, the Lagrange points. Tilt into the sheet view to see the landscape in 3D, then drop test satellites on it and watch what holds.
The five ledges here
What you are looking at
The sheet is the combined pull of both bodies plus the spin of the frame, drawn as height. Both bodies sit in deep wells. L4 and L5 are the two hilltops, and here is the surprise: satellites do not roll off them, because the sideways Coriolis push of the spinning frame steers anything that starts drifting back into a loop. That is why the tadpole orbit works, and why Jupiter has collected thousands of Trojan asteroids on its own L4 and L5 (fourteen of them ride in this catalogue).
L1, L2 and L3 are mountain-pass saddles: a satellite parked there is balanced but not held, like a marble on a ridge. Watch the L1 preset sit quietly for a while and then slide off; real L1 missions burn a whisker of fuel every few weeks to stay put.
The arrive-and-brake preset shows an insertion: a craft screaming past gets one chance to burn. Too little brake and it keeps going; enough and it falls into a loop. Brake into a loose, wide loop and a third thing can happen: the orbit leaks back out through the L1 or L2 pass. The bubble of safety is real and it has doors.
This is the planar circular restricted three-body problem, the classic teaching model. Orbits here are flattened to the plane and the pair is assumed circular. It is the right tool for seeing why these orbits exist, not for flying them.