Integrating gravitational acceleration of particles for simulation

[caspernorth]

I've tried out the two body problem and tried to work out the trajectories with respect to their center of mass frame(located at the origin) as follows (it worked!):

particles = m₁ and m₂
F_m1 = force of m2 on m1
F_m2 = force of m1 on m2

a = Gm₂$\hat{r}$ / r²
$\hat{r}$ is the unit vector
$\hat{r}$ = r/|r|
a _m1 = Gm₂ r / |r|^3
r = √(x^2 + y^2)

and did this for both a1 and a2 but my doubt is this:
why is there a substitution for r vector only when the denominator term stays constant (magnitude).
Wouldn't this imply that particles slows down when r vector decrease?!

at each time intervals shouldn't the magnitude of vector connecting two particles change too? please help.
also why can't type in complex equation forms (a_m1 = [itex]\frac{Gm₂$\hat{r}$ }{r²}[/itex]). it shows the syntax instead of formatted text.

 

[mfb]

why is there a substitution for r vector only when the denominator term stays constant (magnitude).

What do you mean with that?
The denominator stays constant for circular orbits only.

at each time intervals shouldn't the magnitude of vector connecting two particles change too?

In general, it does.

also why can't type in complex equation forms (a_m1 = $\frac{Gm^2\hat{r} }{r^2}$). it shows the syntax instead of formatted text.

Do not nest [itex] and BB-Code tags, that does not work.