Suppose $H(x, y, p)$ is the Hamiltonian associated to $L$ via the Legendre transform. Then:
\begin{align*}
\frac{\partial H}{\partial p} &= y', \qquad \frac{\partial H}{\partial y} = -\frac{\partial L}{\partial y}, \qquad \frac{\partial H}{\partial x} = -\frac{\partial L}{\partial x}.
\end{align*}