Cubic Legendre transform for scalar derivative interactions (${\mathcal{H}}_3(\pi ,\varphi )=-{\mathcal{L}}_3(\varphi ,\dot{\varphi }=\pi )$)

For $\mathcal{L}={\dot{\varphi }}^2/2-{\mathcal{V}}_2+{\mathcal{L}}_3(\varphi ,\dot{\varphi })$, where ${\mathcal{L}}_3$ is cubic in perturbation amplitude, the canonical momentum has $\pi =\dot{\varphi }+O({\varphi }^2)$. Put $\dot{\varphi }=\pi +\Delta$ with $\Delta$ second order. Then $\pi \dot{\varphi }-{\dot{\varphi }}^2/2={\pi }^2/2-{\Delta }^2/2$, 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.