Given the point mass $p_1$ at $(0,0)$ with mass $m_1$ and the point mass at $p_2$ at $(r,0)$ with mass $m_2$ how would you find the positions of $p_1$ or $p_2$ at any time. My first thought was to first solve the problem of what their positions were after some small period of time $\Delta t$. The force by gravity on $p_1$ is $F_1=\frac{Gm_1m_2}{r^2}$ and from the equation $F=ma$ substituting $\frac{v}{t}$ in for $a$ and the $\frac{d}{t}$ for $v$ the result should be $F=\frac{md}{t^2}$ we know the force on $p_1$ and its mass, as well as the time so substituting them results in $\frac{Gm_1m_2}{r^2}=\frac{m_1 d}{\Delta t^2}$ some quick algabra tells me that $p_1$ should have move a distance of $d_1=\frac{Gm_2\Delta t^2}{r^2}$. Taking similar steps brings me to find that $p_2$ travels a distance of $d_2=-\frac{Gm_1\Delta t^2}{r^2}$. From there the same process could be repeated to find the position of of $p_1$ and $p_2$ at the time $2\Delta t$ except by using $r-\frac{G(m_1+m_2)\Delta t^2}{r^2}$ for the new distance between the points. In general if $R(t)$ is the distance between $p_1$ and $p_2$ and if $D_1(t)$ is the distance that $p_1$ moves over the time $\Delta t$ at time the time $t$ then $D_1(t)=\frac{Gm_2\Delta t^2}{R(t)^2}$ and the corresponding function $p_2$ is $D_2(t)=\frac{Gm_1\Delta t^2}{R(t)^2}$. With those two equations $R(t)=\lim_{h\to\infty }\sum_{n=0}^h D_1(\frac{nt}{h})+D_2(\frac{nt}{h})$, but since $D_1$ and $D_2$ are defined that makes are self referential, is there any way out of that? Would the definition $R(t)=\int_0^t\frac{G(m_1+m_2)}{R(u)^2}du$ be equivalent or useful? Most of this work is back of the envelope stuff combined with only really a half understanding of both calculus and Newtonian Mechanics, so any pointers or advice would be greatly appreciated.
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Given the point mass p1 at (0,0) with mass m1 and the point mass at p2 at (r,0) with mass m2 how would you find the positions of p1 or p2 at any time.
This is a difficult problem unless you use a well-known trick: break the problem into the motion of the center-of-mass and the relative motion.
The center-of-mass ($(m_1\vec r_1 + m_2\vec r_2)/(m_1+m_2)$) obeys a very simple equation...
The relative coordinate ($\vec r_1 - \vec r_2$) obeys an equation that you can solve using basic calculus.
$$ \left( \frac{m_1 m_2}{m_1+m_2}\right)\frac{d(\vec r_1 - \vec r_2)}{dt} = -\frac{Gm_1 m_2 (\vec r_1 - \vec r_2)}{|\vec r_1 - \vec r_2|^3} $$
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