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consider object A with mass $m_{A}$ and positional vector $\overrightarrow{r_{A}}$

object B with mass $m_{B}$ and positional vector $\overrightarrow{r_{B}}$

object C with mass $m_{C}$ and positional vector $\overrightarrow{r_{C}}$

since reference frame is inertial (assumed) so

$m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F}{}_{AB}+\overrightarrow{F}{}_{AC}$

$\Rightarrow$$m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})$ ....[1]

for potential energy equation is to be integrated with a positional vector and i am trying hard to figure out what that vector could be, but still no success. please help


Potential energy of a object due to another object $m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}=\overrightarrow{F_{A}}=G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{A}}-\overrightarrow{r_{B}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})$

assuming $\overrightarrow{r_{0}}=\overrightarrow{r_{B}}-\overrightarrow{r_{A}}$

$\Rightarrow m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}=-(1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})$

$\Rightarrow\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{0}}\right).d\overrightarrow{r_{0}}=-\intop_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}\left((1+\frac{m_{A}}{m_{B}})(G\frac{m_{A}m_{B}}{r_{0}^{2}}\hat{r_{0}})\right).d\overrightarrow{r_{0}}$

$\Rightarrow\left.\frac{1}{2}m_{A}v_{0}^{2}\right|_{v_{0}(\overrightarrow{r_{0i}})}^{v_{0}(\overrightarrow{r_{0f}})}=\left.G\frac{m_{A}(m_{A}+m_{B})}{r_{0}}\right|_{\overrightarrow{r_{0i}}}^{\overrightarrow{r_{0f}}}$

So What positional vector should be integrated with equ [1] ? How it is decided ? (i mean what kind of property(s) that vector should possess ?)

EDIT 1

$\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(m_{A}\frac{d^{2}}{dt^{2}}\overrightarrow{r_{A}}\right).d\overrightarrow{r}=\int_{\overrightarrow{r_{i}}}^{\overrightarrow{r_{f}}}\left(G\frac{m_{A}m_{B}}{\left|\overrightarrow{r_{B}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{B}}-\overrightarrow{r_{A}})+G\frac{m_{A}m_{C}}{\left|\overrightarrow{r_{C}}-\overrightarrow{r_{A}}\right|^{3}}(\overrightarrow{r_{C}}-\overrightarrow{r_{A}})\right).d\overrightarrow{r}$ .....[2]

questions

a) what is $\overrightarrow{r}$ in terms of $\overrightarrow{r_{A}}$,$\overrightarrow{r_{B}}$or $\overrightarrow{r_{C}}$ ?

b) solve equ [2].

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I don't seem to be getting what the question is here-you'll have three coupled PDE's for $\vec r_{A}$,$\vec r_{B}$,and $\vec r_{C}$, which you already seem to know how to write down. It turns out that their solution is chaotic, and does not have a closed-form analytic solution.You can certainly use conservation of angular momentum, energy,and linear momentum to reduce the degrees of freedom of the problem, but you will still have remaining coupled differential equations.If you're asking about how to integrate the equations, look into the Euler method: en.wikipedia.org/wiki/Euler_method –  Jerry Schirmer May 11 '11 at 14:39
    
You keep saying "potential energy" but writing equations for forces. I'm pretty sure you mean that you want to integrate the force along a path to get potential energy, but you will have more luck getting a good answer if you are careful with your terminology. –  dmckee May 11 '11 at 14:44
    
@Jerry Schirmer thanks for euler method. but still i dont know answer of question (a) (see EDIT 1). –  tagbw May 11 '11 at 16:38
    
i have gone through many books (and studied few of them) but i have not seen anything like $m_{A}(m_{A}+m_{B})$. am i doing something wrong here (quite sure) or was there some hidden assumption that i did not get. –  tagbw May 12 '11 at 7:20
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1 Answer

What you did is you made a muddled reduction of a two-body system to center of mass motion and a relative coordinate with a reduced mass. The reduced mass of the center of mass is:

$ m_A m_B\over(m_A + m_B)$

and when you divide the Newtonian force $Gm_A m_B/r^2$ by the reduced mass to find the acceleration, you find:

$ |A| = {G(m_A + m_B)\over r^2}$

Then you multiplied this acceleration by $m_A$ to find the motion of one of the objects. This is not appropriate, but it explains your mysterious factor of $m_A (m_A+m_B)$. The references to read are those which discuss reduced mass of a mechanical system.

For your original question, it is best to never integrate to find the potential. The potential is more fundamental than the force, just assume its form a-priori. The reason you are getting confused is because you are integrating with respect to different variables, like relative position, where the accelerations are compounded sums of the accelerations of different particles.

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