Smooth Hamiltonian Systems with Soft Impacts

Abstract

In a Hamiltonian system with impacts (or "billiard with potential"), a point particle moves about the interior of a bounded domain according to a background potential and undergoes elastic collisions at the boundaries. When the background potential is identically zero, this is the hard-wall billiard model. Previous results on smooth billiard models (where the hard-wall boundary is replaced by a steep smooth billiard-like potential) have clarified how a smooth billiard may be rigorously approximated using a hard-wall billiard. These results are extended here to models with smooth background potential satisfying some natural conditions. This generalization is then applied to geometric models of collinear triatomic chemical reactions. (The models are far from integrable n-degree-of-freedom systems with n ≥ 2.) The application demonstrates that the simpler analytical calculations for the hard-wall system may be used to obtain qualitative information with regard to the solution structure of the smooth system and to quantitatively assist in finding solutions of the soft impact system by continuation methods. In particular, stable periodic triatomic configurations are easily located for the smooth highly nonlinear two-and three-degree-of-freedom geometric models.

On Near Integrability of Some Impact Systems

“…The main result here is that under some specific conditions, in a large (O(1) measure) phase space region, smooth nearintegrable dynamics are realized for sufficiently small r and w . Moreover, using [17], it is shown that these results may be extended for the smooth system in which the hard wall is replaced by a soft steep potential, provided the potential is sufficiently steep (notably, the steeper the potential is the larger the perturbation is in the C r+1 topology).…”

“…For physical setups in which bodies at close range experience strong repulsion forces (e.g. the repelling forces between two colliding atoms) [27,30,20,17], a better model for the strong repulsion than the singular hard-wall billiard potential is a smooth steep potential. Hence, consider Hamiltonian systems similar to those discussed above, where the hard billiard is replaced by a smooth potential whose softness is controlled by a small parameter b :…”

“…Traditional examples are near-integrable systems and slow-fast systems [1,2,15]. More recently, analytical tools for studying smooth near-billiard and near-impact systems have been developed [18,30,27,20,17]. In these works, the limit system is a Hamiltonian Impact System (HIS) which describes the dynamics of a particle moving under the influence of a potential inside a domain and reflecting elastically from its boundary.…”

“…By this approach, to better understand systems with very steep potentials at the domain's boundary, one studies the limit system in which the steep part is replaced by impacts. Once the dynamics under the HIS are known, one establishes which of its features persist [27,17] and how those which do not persist bifurcate [30,28].…”

“…Here we provide such a class of prototype impact systems which are near-integrable and are amenable to analysis. Previous near-integrability results for HIS have utilized the local dynamics near periodic orbits [11,4,5,17], near a smooth convex boundary [31,6,5] and near saddle-center homoclinic connection of a quadratic potential with impacts [20]. Another approach utilized the generalized adiabatic theory in 1.5 d.o.f.…”