= Cubic Legendre transform for scalar derivative interactions
{title2=$\mathcal H_3(\pi,\phi)=-\mathcal L_3(\phi,\dot\phi=\pi)$}
For $\mathcal L=\dot\phi^2/2-\mathcal V_2+\mathcal L_3(\phi,\dot\phi)$, where $\mathcal L_3$ is cubic in perturbation amplitude, the <canonical momentum> has $\pi=\dot\phi+O(\phi^2)$. Put $\dot\phi=\pi+\Delta$ with $\Delta$ second order. Then $\pi\dot\phi-\dot\phi^2/2=\pi^2/2-\Delta^2/2$, so through cubic order the <Legendre transform in mechanics> gives $\mathcal H_3=-\mathcal L_3$ evaluated with the free momentum. At fourth order there are extra terms, so derivative interactions do not obey this simple rule at all orders.
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