MULTIBUMP ORBITS NEAR THE ANTI-INTEGRABLE LIMIT FOR LAGRANGIAN SYSTEMS

We consider Lagrangian systems with Lagrangians L(q, epsilon(q) overdo t, t, epsilon), depending slowly on time. In the limit epsilon --> 0 ( adiabatic limit) the system becomes autonomous with Lagrangian L(q, q' , t, 0) depending on the parameter t. By using variational methods, fo r small epsilon not equal 0, we construct trajectories that are close to chains of homoclinic orbits of the limit system. This is a generali zation of a result of Cherry, who considered the one-dimensional nonde generate case. Some multidimensional nondegenerate cases were studied by Palmer. The trajectories we construct are similar to the trajectori es of symplectic maps in the so-called anti-integrable limit.