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E92M3
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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?
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?
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