Finding r(ϕ) from r(t) and ϕ(t) to plot Newtonian orbits

[Poirot]

Homework Statement

I'm trying to plot Newtonian orbits in Mathematica with the goal of extending this to orbits including GR after. I've derived equations for r(t) and $\phi$(t) (see below) by integrating $\dot{r}$ and $\dot{\phi}$ (see below also) from E = 1/2mv² +V(r) with velocity being split into angular and radial terms and V(r) = GMm/r. I think I then need to find an equation for r($\phi$) to then plot in Mathematica using the ParametricPlot function but I can't see how to do this as my equations for r(t) and $\phi$(t) are tricky.

Homework Equations

$\dot{r} = \sqrt{\frac2m(E-\frac{GMm}{r})-\frac{L^2}{m^2r^2}}$
$\dot{\phi} = \frac{L}{mr(t)^2}$
$r(t)=\sqrt{\frac2m(E-\frac{GMm}{r})-\frac{L^2}{m^2r^2}} t + r_0$
$\phi(t)=\frac{L}{m}\frac{ln(m\sqrt{\frac2m(E-\frac{GMm}{r})-\frac{L^2}{m^2r^2}} t + r_0m)}{\sqrt{\frac2m(E-\frac{GMm}{r})-\frac{L^2}{m^2r^2}}} + \phi_0$
$r(0)=r_0$
$\phi(0)=\frac{L}{m}\frac{ln(r_0m^2)}{\sqrt{\frac2m(E-\frac{GMm}{r})-\frac{L^2}{m^2r^2}}} + \phi_0$

The Attempt at a Solution

I tried finding initial conditions at t=0 (see equations above) but I'm not sure if any of this is even right? It seems to have gotten to complicated. If everything I have done so far is correct, I'm not sure how to then find r($\phi$) and if I was to find this how to convey all this information into a ParametricPlot function in order to obtain the orbits.

Any help would be greatly appreciated, Thanks.

 

[vela]

If you have $r(\phi)$, you could use PolarPlot. For ParametricPlot, you need expressions for $x(t)$ and $y(t)$. With $r(t)$ and $\phi(t)$, that's straightforward using the usual polar-to-Cartesian coordinate transformation.

 

[Poirot]

Thanks for the reply!
I had planned to find $r(\phi)$ and then used ParametricPlot with $r(\phi)Cos(\phi)$ and $r(\phi)Sin(\phi)$ so that I could plot independent of t (As this is what I think my supervisor implied to do) but in practice I've gotten myself quite confused. I tried plotting $r(t)$ and $\phi(t)$ as a ParametricPlot of t also but this didn't seem to work, which I think I can see why from what you've just said. So I've just retried this using $r(t)Cos(\phi(t))$ and $r(t)Sin(\phi(t))$ plotting with changing t but this didn't seem to work either? It may be that I'm just not that great at Mathematica too, but I can't see any clear mistake I've made in syntax.

Thanks again.