Thanks for that, those are some pretty big mistakes. They are vectors as that's how they are in the book.
About the 3rd term in the potential, I agree it would be needed if it was the potential was for the system but as far as I understand I'm only interested in the potentials that act upon m3? Just as I have excluded the kinetic energies of m2 and m1, I exclude the potential that acts between them? Let me know if I'm wrong.
I'll add the third term for now though as I can always change it later
So I'm assuming this is right apart from whether the 3rd term in the potential should be included or not.
Then the following transformation is applied to make to coordinate system rotate at the same frequency.
Then thetadot in the equation for L is changed and expanded giving
Then the book says switch to cylindrical coordinates I'll quote here:
we can write the Lagrangian in terms of the rotating system by using theta' = theta + wt as the transformation to the rotating frame. Thus, the Lagrangian in the rotating coordinates can be written in terms of the cylindrical coordinates, rho, theta and z, with rho being the distance from the center of mass and theta the counterclockwise angle from the line joining the two masses
Thus:

Which is exactly the same, just changing r to rho and t to z?
In the standard cylindrical coordinates, z = z. So why does the t vanish and become a z in the book? z is not just another variable for the time either as they add in the zdot^2 term in the kinetic energy implying they mean the kinetic energy in the z direction. So how did my time dependent potential turn into a z dependent potential? Especially since it is a twodimensional problem