Light ray paths near schwarzschild blackhole

[George Jones]

That would be

\frac{d^2r}{d\lambda^2} = - \frac{L^2(3M-r)}{r^4}

Which gives me the closest radius for a stable orbit of a photon as 3M

Reduce this second-order equation to two first-order equations by setting p = dr/d\lambda, so that dp/dt = d^2r/d\lambda^2. The set of equations that describes the worldline of a photon then is

<br /> \begin{equation*}<br /> \begin{split}<br /> \frac{d \phi}{d \lambda} &amp;= \frac{L}{r^2} \\<br /> \frac{dr}{d\lambda} &amp;= p \\<br /> \frac{dp}{d \lambda} &amp;= \frac{L^2(r - 3M)}{r^4}.\\<br /> \end{split}<br /> \end{equation*}<br />

Assuming that all required values are given, write a few lines of (pseudo)code that uses the simplest, most intuitive method (Euler's method) to solve these equations.

 

[nocks]

Having a bit of trouble with this but I have p as

p = \frac{L^2(2M-r)}{2r^3}

 

[nocks]

wow it's been a while since I looked at this project.
Thought i'd come back to it :)

Reduce this second-order equation to two first-order equations by setting p = dr/d\lambda, so that dp/dt = d^2r/d\lambda^2. The set of equations that describes the worldline of a photon then is

<br /> \begin{equation*}<br /> \begin{split}<br /> \frac{d \phi}{d \lambda} &amp;= \frac{L}{r^2} \\<br /> \frac{dr}{d\lambda} &amp;= p \\<br /> \frac{dp}{d \lambda} &amp;= \frac{L^2(r - 3M)}{r^4}.\\<br /> \end{split}<br /> \end{equation*}<br />

Assuming that all required values are given, write a few lines of (pseudo)code that uses the simplest, most intuitive method (Euler's method) to solve these equations.

A few questions. Since I lack a good maths background, I would appreciate some advice on how I would implement these 3 equations into a numerical solver like Euler's or Runge-Kutta to get back some results that I could plot.
e.g. Do I pre-define L?, how do I know which direction the photon is travelling?

currently trying to use this : http://www.ee.ucl.ac.uk/~mflanaga/java/RungeKutta.html