potential energy of an object due to other two objects

Question

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].

Answer 1

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.