I would like to express the potential of 2 orbiting objects in the rotating frame, but I'm not quite doing it right. I am a physics major but since I took AP, my mechanics is quite bad. Here's what I'm doing, please tell me what am I missing.

First, I consider two objects denoted with 1 and 2 with circular orbits around their common center of mass. When I in the rotating frame where the origin is the common center of mass, objects 1 and 2 are both on the y axis at a and -b respectively. I don't know how to write the potential for the Coriolis and centrifugal terms; however, I can indeed write the acceleration and thus the force. I would like to eventually recover the potential through:

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If I now consider a mass m at an arbitary point, I can write the following:

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Dividing by the mass m on both sides I get:

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The last term is zero since there is no change in omega, namely the angular velocity. The centrifugal term can also be simplified. Since the vector r always points outwards nd the angular velocity always points in the z-direction. After taking care of the minus sign and defining r in terms of x and y I get:

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Now here are my problems:

How can I handle
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How can I write it it terms of x and y?

Then to find V(x,y), can I just do:

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If this is correct then how do I get the constant that should fall out when I do the integral? As I on the right track? All I am trying to to is to look at the stability at the Lagrange points. Should I pursuit another path? Is there another way to obtain the potential?