An analytic counterexample to the KAM theorem

Abstract

We study the non-existence of KAM tori for quasi-integrable, analytic Lagrangians. Let ${\cal L}\colon {\bf T}^m\times{\bf R}^m \to {\bf R}$, ${\cal L} (Q,\dot{Q})=\tfrac{1}{2}\vert\dot{Q}\vert^2+h(Q)$ and let $\bar\omega\in{\bf R}^m$ be a frequency exponentially close to resonances. We find $h$ analytic of norm arbitrarily small such that ${\cal L}$ has no invariant torus of frequency $\bar\omega$ projecting diffeomorphically on $\T^m$