Abstract: We consider the problem whether small perturbations of integrable mechanical systems
can have very large effects.
It is known that in many cases, the effects 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 carefully 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