Quasiperiodic Motion in the Hamiltonian Systems of the Billiard Type

Abstract

It is shown that two-degree-of-freedom Hamiltonian systems of the billiard type are equivalent to adiabatically varying one-degree-of-freedom Hamiltonian systems for solutions staying near the boundary. Under some nondegeneracy conditions such systems possess a large set of quasiperiodic solutions filling out two-dimensional invariant tori. The latter separate the extended phase space into layers providing stability for all time. The result is illustrated on a few examples. [S0031-9007(98)07815-6] PACS numbers: 05.45. + b, 02.30.Hq, 03.20. + i Billiards have been frequently used to investigate the relation between the classical and quantum mechanics. This relation is clear for exactly solvable systems but is less understood if the classical system has both regular and chaotic components in the phase space. A nonelliptic convex billiard provides one of the simplest examples of such systems. Indeed, since in billiards the particle interacts only with a one-dimensional boundary and is free in the interior, it makes analysis and numerical simulations easier. On the other hand, a convex billiard is known to possess a large family of caustics (defined as the envelopes of light ray trajectories) accumulating at the boundary [1]. The caustics represent invariant sets in the phase space and, thus, are important for constructing the quasimodes and estimating the corresponding eigenvalues for the quantum billiard [2,3].Such near integrable behavior in the vicinity of the boundary of a convex billiard can be anticipated by noting that a trajectory nearly tangent to the boundary will experience many collisions with the boundary before the curvature will significantly change. This raises a hope of introducing different time scales and obtaining an adiabatic invariant.An adiabatic invariant, which is given by the action variable, is well known to exist in slowly varying smooth Hamiltonian systems H͑q, p, l͒, where l et [4,5]. Moreover, it was shown by Arnold [6] that in a nonlinear system with a slowly periodically varying smooth Hamiltonian the adiabatic invariant is conserved perpetually under some nondegeneracy conditions. The proof is based on the application of Kolmgorov-Arnold-Moser (KAM) theory to a reduced Hamiltonian obtained from the original Hamiltonian H͑q, p, l͒ after a series of canonical transformations are carried out. This result showed that the majority of slowly periodically varying smooth Hamiltonian systems possess a large set of invariant tori carrying quasiperiodic motion.Contrary to the smooth case, the stability problem for the systems with impacts had to be resolved on a case by case basis, see [2,[7][8][9][10][11], by first obtaining a "bouncing map" and then bringing it to the near integrable form.In this Letter, we use the adiabatic invariant approach to establish a counterpart of Arnold's result for twodimensional billiard systems (or equivalently for slowly periodically varying one-dimensional impact oscillators) providing a unified approach to investigate the regularity of motion in the system...

Chaotic dynamics of thermal atoms in labyrinths created by optical lattices

“…Since the detuning is large, dissipative effects are expected to be small; thus, in the high-intensity regime, the system dynamics is very similar to that of two-dimensional billiards with a topology determined by the parabolic symmetry of the light beam. It is known [25] that this kind of dynamical system has a large set of quasiperiodic solutions. We consider this the key to understanding the saturation effect illustrated in figure 2.…”