Title:

Antiintegrability in Lagrangian systems

Three examples of application of the antiintegrability concept in Lagrangian systems are proved, concerning the continuation of a class of trajectories from the antiintegrable limit. All three examples were proposed by Robert S. MacKay. The first example arises in adiabatically perturbed systems. With an assumption that the adiabatic PoincaréMelnikov function has simple zeros, we constructed a variational functional whose critical points give rise to a sequence of homoclinic trajectories for the unperturbed Lagrangian in the adiabatic limit but a sequence of multibump trajectories under perturbations. We found there is a compact set, which is a Cantor set, such that the Poincaré map induced by the phase flow restricting to it is conjugate to the Bernoulli shift, in our case, with three symbols. Hence the approach of the antiintegrability to the adiabatically perturbed problems is equivalent to the one which combines the PoincaréMelnikov method and the BirkhoffSmale theory. The second example occurs in the Sinai billiard system. The antiintegrable limit is the limit when the radius of the scattererdisc goes down to zero, and the system becomes "δbilliards". The orbits of the δbilliards are the antiintegrable orbits which are piecewise straight lines joining zeroradius discs to discs, and are easily obtained. Under some nondegeneracy conditions, we proved all antiintegrable orbits can be continued to the small radius case, and found that any periodic orbit has infinitely many homoclinic orbits as well as heteroclinic orbits to any others. These exists a compact set, which is also a Cantor set, such that the billiard map restricted to it is conjugate to a subshift of finite type with an arbitrarily given number of symbols. We studied in the third example when the scatterers are approximated by repulsive potentials such as the Coulomb potential ε/r, where ε and r are nonnegative numbers and r is the distance from the potential centre. In the Coulomb potential case, the antiintegrable limit is the ε → 0, and the system becomes the δbilliard system. Then we found that the results in the Sinai billiards also hold here when ε > 0 but small. More general type of repulsive potentials were also investigated and a sufficient condition under which antiintegrable trajectories persist was given.
