MTW Gravitation Ex. 25.16: Comparing Equations & Expressing u_0 in \tilde{L}

[luinthoron]

Hello, I am trying to work out this exercise for my personal research connected with my bachelor thesis. The task is to compare equations (25.42) and (25.47) and express $u_0$ in terms of \tilde{L}. I have so far put the two equations together getting

\begin{equation}
12u^2u_0\tilde{L}^2-18uu_0^2\tilde{L}^2-u_0^2\tilde{L}^2+2uu_0\tilde{L}^2-\tilde{E}_0^2=2u-1
\end{equation}

After this I tried putting some terms together but I think I am missing another equation since there are in fact two unknowns: $u_0$ and $\tilde{E}_0$ or is there some trick I am missing?

For those without access to MTW, here are the equations: \\

(25.42)
\begin{equation}
\left(\frac{\mathrm{d}u}{\mathrm{d}\varphi}\right)^2=\frac{\tilde{E}^2}{\tilde{L}^2}-\frac{1}{\tilde{L}^2}\left(1-2u\right)\left(1+\tilde{L}^2u^2\right)
\end{equation}

and (25.47)

\begin{equation}
\left(\frac{\mathrm{d}u}{\mathrm{d}\varphi}\right)^2+\left(1-6u_0\right)\left(u-u_0\right)^2-2\left(u-u_0\right)^3=\frac{\tilde{E}^2-\tilde{E}_0^2}{\tilde{L}^2}
\end{equation}

Thank you.

 

[Bill_K]

Each equation expresses (du/dφ)² in terms of a cubic in u. Expand out the cubics and equate the coefficients of each power of u.

 

[luinthoron]

Thank you for the advice. I ended up with:

\begin{equation}
\tilde{E}_0^2=4\tilde{L}^2u_0^3-\tilde{L}^2u_0^2+1
\end{equation}

\begin{equation}
3u_0^2-u_0=-\frac{1}{\tilde{L}^2}
\end{equation}

The second one is a quadratic equation, so I can write the solution

\begin{equation}
{u_0}_{1,2}=\frac{1\pm\sqrt{1-\frac{12}{\tilde{L}^2}}}{6} .
\end{equation}

But this would give a condition for \tilde{L}, which I find suspicious. Also which of the two roots is correct? It stays possitive in both cases and I can't come with any other clue to help me choose. Any additional hints, please?