I'm new here at the physics site, and not really that deep into the area of which i'm going to ask a question about now. Therefore please feel free to ask clarifying questions.
I'm trying to deal with the Lagrangian Points in the S-E-system. I'm at the moment just going through a paper in which a derivation of the coordinates (cartesian) of the points is done.
Now, what I have understood so far: We have the 3 bodies with masses $M_1$, $M_2$ and $m$. The distance of $M_1$, $M_2$ and $m$ are $\vec{r_1}$, $\vec{r_2}$ and $\vec{r}$, respectively. The gravitational force excerted on mass $m$ can be written $$\vec{F}=-G \frac{M_1m}{|\vec{r}-\vec{r_1}|^3}(\vec{r}-\vec{r_1})-G \frac{M_2m}{|\vec{r}-\vec{r_2}|^3}(\vec{r}-\vec{r_2})$$ And so we have the equation of motion $$\vec{F}(t)=m\frac{d^2\vec{r}(t)}{dt^2}$$ Which we want to solve, so that the distances between the 3 bodies are fixed.
Now, this paper makes as a postulate that adopting a co-rotating frame of reference is covenient. I follow, and the effective force is found - which is relative to the angular velocity $\omega$ and the inertial force $F$: $$\vec{F_\omega}=\vec{F}-2m(\vec{\omega}\times\frac{d\vec{r}}{dt})-m\vec{\omega}\times(\vec{\omega}\times\vec{r})$$ It then states: "We can then write the general potential energy of the system as:" $${U_\omega}=U-\vec{v}\cdot(\vec{\omega}\times\vec{r})+[(\frac{\vec{\omega}\times\vec{r}}{2})\cdot(\vec{\omega}\times\vec{r})]$$ And then at last (the last of what i don't understand at the moment), it states that the effective force can again be derived from the generalized potential: $$\vec{F_\omega}=-\nabla_\vec{r}U_\vec{\omega}+\frac{d}{dt}(\nabla_\vec{v}U_\vec{\omega})$$
Now, first of all, i'm not really familiar with Lagrangian mechanics, but have read a bit. Is the $\omega$-index on the force showing me that we are talking about generalized forces? I'm not quite sure. Next, from where does this expression of the potential come from? Can someone refer to it, or show how it is derived? At last, I believe the derivation of the second expression of the effective force comes from the equation $$Q_j=-\frac{\partial U}{\partial q_j}+\frac{d}{dt}(\frac{\partial U}{\partial \dot{q_j}})$$ I would like to know, if i'm right about this, and also, if someone has the time, see the derivation of this.
I hope my questions are understandable, and I hope some of you can answer them and explain some of the things to me.
