Solving 1D First Order Equations for 3D Mass Positions and Velocities

[Blanchdog]

Homework Statement

Use the given equations to obtain the first order differential equations for the system of two gravitating bodies. Write them down on paper in terms of the individual components of the motion.

Relevant Equations

m1 r'' = -G m1 m2 (r1 - r2)/|r1 - r2|^3
m2 r'' = -G m1 m2 (r2 1 r1)/|r2 - r1|^3

Okay so I need to find 12 one dimensional first order equations that describe the position and velocity of both masses in 3 dimensions. The equations for the second body will be easy once I figure out how to do the first body, so I'll ignore that for now. For the first equation, I can rearrange it to become:

r'' = -G m2 (r1 - r2)/|r1 - r2|^3

And I can break that down into two first order equations

a1' = a2
a2' = -G m2 (a1 - r2)/|a1 - r2|^3

I'm just stuck on how I now break those up into x y and z and then solve them symbolically to be able to write them down.

 

[anuttarasammyak]

Hi. You may find changing coordinates from $r_1,r_2$ to $r_G,r$ ,where $r_G$ is the coordinate of center of mass and $\mathbf{r}=\mathbf{r_2-r_1}$, will be helpful.
$$\dot{x}_G=const.$$
$$\ddot{\mathbf{r}}=-G(m_1+m_2)\frac{\mathbf{r}}{r^3}$$