Questions regarding the relative motion of three masses.

Usually, the trajectory of a spacecraft is modelled as if it is orbiting just one body at a time, using the patched-conics approximation based upon the well known two-body equations. This is however not strictly true, and the errors might be noticeable at the edge of a planets sphere of influence, or during a fly by manoeuvre. Furthermore, the concept of Lagrangian points does not exist in the two-body model.

Simplifications often made in the three-body include:

Mass of third body is negligible
The two primaries orbit in a circular barycentric orbit
Orbits are co-planar

This is known as the circularly restricted three body problem (CRTBP or CR3BP)