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This paper considers five examples of hamiltonian systems for which the existence of Arnold's mechanism for diffusion has been shown. These systems have in common that each of them is the perturbation that couples a number of rotators to a pendulum. The main result is that, for all systems considered and for all suffiently small values of the perturbation paramenter, there are orbits whose action variables have a drift of order one in a time which is inversely proportional to the splitting of the homoclinic whiskers. The paper reviews the necessary results concerning the Hamiltonian and uses them to apply Mather theory to the estimate on the drift. These results are important in veryfying the optimality of Nekhoroshev-type theorems

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We consider the problem − u + F u x u = 0 on R n , where F is a smooth function periodic of period 1 in all its variables. We show that, under suitable hypotheses on F , this problem has a family of non-self-intersecting solutions u D , which are at finite distance from a plane of slope 0 0 with irrational. These solutions depend on a real parameter D; if D = D , then the closures of the integer translates of u D and of u D do not intersect.

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 $\T^m$