English: Energy potential analysis of the restricted three-body problem. The first equation shows that the total potential (U_tot) is a combination of the gravitational potential of the two primary bodies along with the centrifugal effect of the rotating reference frame (which is an inertial effect of the third body in a non-rotating frame). This relationship can be manipulated into the second equation, which more clearly shows the centrifugal component to be of paraboloid geometry, as illustrated in the bottom-middle graph. The final resulting surface has Lagrange points located where the gradient is zero, as indicated by the blue lines on the final graph.

A force vector field has a one-to-one correspondence with these potential surfaces, as is specifically explained in this image:
Force Field-Gravity Well Inter-relationship.png

Note 1: Half of the field has been cut away for clarity. Also, the mass parameter (mu) being graphed is 0.25. This is a significant difference from the Earth-Moon system, so the labels are only qualitatively representative. There are also Coriolis effects which are dynamic and not shown.

Many science museums have "gravity well" demonstrations. A similar display can be created for three-body orbits by using a 3-d printer to make the combined gravity potential surface, and then rotating it at a scaled rate. A steel ball bearing could then be placed at the stable L4 or L5 locations (equilateral to the primary masses) and if the model is properly constructed then these ball bearings will stay in place (until air resistance drags them away). The centrifugal and Coriolis effects are simply manifestations of inertial effects when observed from the rotating reference frame.

Note 2: Because the rotation rate of the primary masses is a function of gravity, it should be understood that the labeling of factors as being due to "gravity" and due to "rotation" does not mean that the latter has nothing to do with the former. This should be clear upon inspection of both equations. If these labels were not distilled down to a single word for the sake of simplicity, the long-hand labels could be: "potential due to the force due to gravity" (gravity) and "potential due to the force due to the rotation due to the force due to gravity" (rotation).

Graphics generated by HiQ. PDF image edited with GIMP and MS Paint.