- #1

Digital Honeycomb

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## Homework Statement

From the homework:

In General Relativity it is found that the radial equation of an object orbiting a non-rotating black hole has the form $$\dot r^2 + (1 - 2 \frac {V_o} {r} ) (\frac {l^2} {r^2} + 1) = E^2$$ where ##r## is the radial coordinate, ##l## is the angular momentum, and ##E## is the total energy; the potential ##V_o = GM_o##, where ##M_o## is the mass of the black hole. Show, using the standard substitution ##u = \frac {1} {r}##, and rewriting ##\dot r## in terms of ##\dot \phi## and ##u' = \frac {du} {d\phi}##, that we can write a differential equation for ##u(\phi)## of form $$u'' + u - \frac {V_o} {l^2} - 3V_ou^2 = E^2$$

## Homework Equations

All shown above.

## The Attempt at a Solution

First off I'm not entirely sure how to rewrite ##\dot r## in terms of ##\dot \phi##. In the notes it is described that the angular momentum ##l = mr^2\dot\phi##, so therefore this equation could be written in the form $$\dot r = \sqrt{E^2 - (1 - 2V_o u)(m\dot\phi +1)}$$ but it is not exactly like I can plug this back into the original equation. I can also try taking that since ##u = \frac {1} {r}##, that ##\frac {du} {dt} = \frac {-1} {r^2} \frac {dr} {dt} = \frac {du} {d\phi} \frac {d\phi} {dt}## and so ##\dot r = \frac {-1} {u^2} u' \dot \phi## but putting this into the first equation, I'm not exactly sure where the ##u''## is supposed to come from. Am I coming at this from the wrong direction entirely? Thank you!