Deriving equations of motion for restricted 3-body problem

[tomwilliam2]

My textbook is describing a 3-body situation where there are two large masses rotating around their barycentre, and a third much smaller mass experiencing gravitational forces from the two larger masses. If the frame of reference is the one in which the barycentre is at rest, then it is not inertial. For the smaller mass, there is an angular speed $\Omega$ to take into account.
The gravitational force due to the two masses is clear, but when my textbook sets out the equation of motion for this mass, it just gives the result in this frame of reference as:
$$\mathbf{\ddot{r}} + 2(\mathbf{\Omega \times \cdot{r}})+(\mathbf{\Omega \cdot r})\mathbf{\Omega - \Omega^2 r}=-G\left(\frac{M_1}{r_1^3}\mathbf{r_1} + \frac{M_2}{r_2^3}\mathbf{r_2}\right) + \frac{1}{m}\mathbf{F_d}$$
Without any explanation. I'd like to try and derive it for myself. Any idea how to go about doing that?
Obviously I could start with:
$$\mathbf{\ddot{r}} + \mathbf{\alpha(\Omega)}=-G\left(\frac{M_1}{r_1^3}\mathbf{r_1} + \frac{M_2}{r_2^3}\mathbf{r_2}\right) + \frac{1}{m}\mathbf{F_d}$$
Where $\alpha$ represents the contribution to the acceleration caused by the rotating frame of reference, and $F_d$ is the disturbance forces.

 

[Orodruin]

If the frame of reference is the one in which the barycentre is at rest, then it is not inertial

This is not correct. There is a perfectly valid inertial frame where the barycenter is at rest. The point is that you want to go to the rotating frame in which the larger bodies are also at rest.

You can derive the result directly by considering a set of basis vectors that are rotating with the system. Note that the vector $\ddot{\bf r}$ as probably defined in your text only contains the time derivatives of the components, not the entire vector including the basis. Newton's second law holds for the entire vector - including the basis vector.