Abstract. 
We consider the problem whether small perturbations of integable
mechanical systems can have very large effects. It is known that in
many cases, the effcts of the perturbations average out, but there are
exceptional cases (resonances) where the perturbations do accumulate.
It is a complicated problem whether this can keep on happening because
once the instability accumulates, the system moves out of resonance.
V. Arnold discovered in 1964 some geometric structures that lead to
accumulation in carefuly constructed examples. We will present some
other geometric structures that lead to the same effect in more
general systems and that can be verified in concrete systems. In
particular, we will present an application to the restricted 3 body
problem. We show that, given some conditions, for all sufficiently
small (but non-zero) values of the eccentricity, there are orbits near
a Lagrange point that gain a fixed amount of energy. These conditions
(amount to the non-vanishing of an integral) are verified numerically.
Joint work with M. Capinski, M. Gidea, T. M-Seara.