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## Main Question or Discussion Point

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.

\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.