American Institute of Mathematical Sciences

$\qquad\qquad \qquad\qquad L_{\epsilon,\mu}(q,\dot{q},Q,\dot{Q},t) = \frac{\dot{Q}^2}2+\frac{\dot{q}^2}2 + \epsilon(1-\cos q)+ \mu f(q,\dot{q},Q,\dot{Q},t,\mu) $

which has been studied by many authors in connection with Arnold's diffusion. Extending [2] prove, by variational means, that, for suitable perturbations including for example:

$\qquad\qquad \qquad\qquad f(q,\dot{q},Q,\dot{Q}, t,\mu)=(1-\cos q)(\cos Q+\cos t) + \mu^{p-1} \sin(q+Q) \quad (p>2) $

if $\mu$ is small enough, exists a diffusion orbit of $L_{\epsilon, \mu}$ such that $\dot{Q}(t)$ undergoes a variation of order $1$ in a time $t_d$ polinomial in $\mu$, $t_d\approx \frac 1{\mu^2}$.