\section{}

Starting from (6), we wish to calculate the minimum action for a system governed by the effective lagrangian:

\begin{equation}
L_{FW}=\frac{1}{2}\left(\frac{\dot{x}-f(x)}{g(x)}\right)^2.
\end{equation}

By doing a legendre transformation we can go to the associated hamiltonian form:

\begin{equation}
H_{FW}(p,x)=p\dot{x}-L_{FW}=p\left(p\frac{g^2(x)}{2}+f(x)\right).
\end{equation}

Since the lagrangian does not depend explicitely on time, the Hamilton-Jacobi equation for the problem reduces to the simple form $H_{FW}(\partial_xS_{FW},x)=0$, where $S_{FW}$ is the action for the problem. This leads to the differential equation for $S_{FW}$:

\begin{equation}
\partial_xS_{FW}=-2\frac{f(x)}{g^2(x)},
\end{equation}
shown in equation (7) of the text.
\end{document}
