Variational destruction of invariant circles

We construct a sequence of generating functions $(h_n)_{n\in\N}$, arbitrarily close to an integrable system in the $C^r$ topology with $r<4$ for $n$ large enough. With the variational method, we prove that for a given rotation number $\omega$ and $n$ large enough, the exact monotone area-preserving twist maps generated by $(h_n)_{n\in\N}$ admit no invariant circles with rotation number $\omega$.