integrable system can be locally written as a function $H_0({\bf A})$ of the action variables ${\bf A}=(A_1,A_2,\ldots,A_d)\in{\mathbb R}^d$, i.e. it does not depend on the conjugate angle variables ${\bm\theta}=(\theta_1,\theta_2,\ldots,\theta_d)\in{\mathbb T}^d
={\mathbb R}^d/(2\pi {\mathbb Z})^d$, where ${\mathbb T}^d$ is the $d$-dimensional torus parameterized by $[0,2\pi[^d$. The equations of motion for $H_0({\bf A})$ show that the action variables are constant, and consequently the trajectories with given actions $\bf{A}$ are confined to evolve on a $d$-dimensional torus with frequency vector ${\bm \omega}_0({\bf A})= \partial
H_0/\partial {\bf A}$. The dynamics on this torus is periodic or quasi-periodic: ${\bm \theta}(t)={\bm \omega}_0({\bf A}) t+{\bm
\theta}(0)$ with frequency vector ${\bm \omega}_0({\bf A})$. In the particular case given by Hamiltonian (\[Ham1\]) an integrable situation is given by $\varepsilon_1=\varepsilon_2=0$ so that the dynamics of the integrable system $H=p^2/2$ is characterized by constant velocity ($p=const$). A monokinetic beam of charged particles remains monokinetic.

If the system described by $H_0$ is perturbed, i.e. we consider the Hamiltonian $$\label{eqn:Hgene} H({\bf A},{\bm \theta})=H_0({\bf A})+V({\bf
A},{\bm \theta})\nonumber,$$ the integrability is generically lost and the system becomes chaotic. Even if KAM theorem establishes the stability with respect to small perturbations of invariant tori with a sufficiently incommensurate frequency vector these tori are destroyed when the 