Anti-integrability in Lagrangian systems

• Main Author: Chen, Y-C.
• Published: University of Cambridge 2002
• Subjects: 519
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597512

Three examples of application of the anti-integrability concept in Lagrangian systems are proved, concerning the continuation of a class of trajectories from the anti-integrable limit. All three examples were proposed by Robert S. MacKay. The first example arises in adiabatically perturbed systems. With an assumption that the adiabatic Poincaré-Melnikov 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 multi-bump 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 anti-integrability to the adiabatically perturbed problems is equivalent to the one which combines the Poincaré-Melnikov method and the Birkhoff-Smale theory. The second example occurs in the Sinai billiard system. The anti-integrable limit is the limit when the radius of the scatterer-disc goes down to zero, and the system becomes "δ-billiards". The orbits of the δ-billiards are the anti-integrable orbits which are piecewise straight lines joining zero-radius discs to discs, and are easily obtained. Under some non-degeneracy conditions, we proved all anti-integrable 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/<I>r</I>, where ε and <I>r </I>are non-negative numbers and <I>r</I> is the distance from the potential centre. In the Coulomb potential case, the anti-integrable limit is the ε → 0, and the system becomes the δ-billiard 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 anti-integrable trajectories persist was given.