Potential of 2 Orbiting Objects in Rotating Frame

[E92M3]

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 accleration and thus the force. I would like to eventually recover the potential through:

$$\vec{F}(x,y)=-\nabla V(x,y)$$

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

$$\vec{F_{net}}=\vec{F_{g,1}}+\vec{F_{g,2}}+\vec{F_{Coriolis}}+\vec{F_{centrifugal}}+\vec{F_{azimuthal}}$$

Dividing by the mass m on both sides I get:

$$\vec{a}(x,y)=\frac{-Gm_1}{x^2+(y-a)^2}-\frac{Gm_2}{x^2+(y+b)^2}-\vec{\omega}\times (\vec{\omega}\times \vec{r})-2\vec{\omega}\times \dot{\vec{r}}+\dot{\vec{\omega}}\times \vec{r}$$

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:

$$\vec{a}(x,y)=\frac{-Gm_1}{x^2+(y-a)^2}-\frac{Gm_2}{x^2+(y+b)^2}+\omega^2\sqrt{x^2+y^2}-2\vec{\omega}\times \dot{\vec{r}}$$

Now here are my problems:

How can I handle
$$\dot{\vec{r}}$$

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

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

$$V(x,y)=\int\int\vec{F}(x,y)dxdy$$

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 Lagarange points. Should I persuit another path? Is there another way to obtain the potential?

 

[mikeph]

There are other ways to investigate the stability that I know- instead of finding the potential field, you can write the equations of motion in a non-rotating frame for the third body, and then perform a coordinate transformation which (like you say) keeps the two massive bodies on the axis. After some manipulation all the explicit time dependences fall out (though time derivatives are still abundant). With this equation you can performs a typical stability analysis, by replacing the position with the known stationary solution (equilateral triangles, right?), plus a small perturbation. You can then expand these in your equations of motion in the rotating frame, and after a (considerable) amount of calculation, you will be able to factor the resulting equation and find the limit of stability, where some roots go imaginary.I write all this because I did the exact same thing for my dissertation last year, and did quite well, but I did not use the method you are using with regards to energy. By the way, your final velocity term cannot be simplified as it is the Coriolis force! It does not affect the stationary points but will affect the stability of them.

In the end, you should find that for some mass ratios (of the two big bodies), the perturbations oscillate (with different periods in different directions, leading to some cool shapes), but after a certain critical ratio stability breaks down. But I am not saying that this is the wrong route, just that there is a pure algebraic route that I know works neatly. I warn you: it took me a few days to work through, you might want to find a computer package to do some of the more complicated transformations.

I'm pretty interested in this stuff so let me know if you need any more help- but bear in mind I did no energy analysis at all, and didn't focus on potential fields at all (though some treatment exists on a page I remember, http://map.gsfc.nasa.gov/media/ContentMedia/lagrange.pdf)

 

[E92M3]

The link was very useful indeed, but there's still something I don'quite get. How did they come up with equation 5?
$$U=U_g-\frac{d\vec{r}}{dt}(\vec{\omega}\times\vec{r})+\frac{1}{2}(\vec{\omega}\times\vec{r})^2$$

1. Where did this come from?
2. What happened to the small "m" which is required to get units of energy?
3. Why does it differ from this classical mechanics text http://books.google.com/books?id=1k...ffective potential of coriolis force&f=false"? Even if you say that the coriolis term is included in V, why does the sign of the centrifugal term differ?