On near integrability of some impact systems

TL;DR

This paper demonstrates that certain Hamiltonian impact systems with a straight wall exhibit smooth near integrable behavior under small perturbations, extending the class of known near-integrable impact systems.

Contribution

It introduces a new class of impact systems showing near integrability, including explicit formulas and robustness under boundary and potential perturbations.

Findings

Near integrability persists away from singularities.

Explicit formulas for the near integrable return map are derived.

Near integrability extends to smooth steep potentials replacing hard boundaries.

Abstract

A class of Hamiltonian impact systems exhibiting smooth near integrable behavior is presented. The underlying unperturbed model investigated is an integrable, separable, 2 degrees of freedom mechanical impact system with effectively bounded energy level sets and a single straight wall which preserves the separable structure. Singularities in the system appear either as trajectories with tangent impacts or as singularities in the underlying Hamiltonian structure (e.g. separatrices). It is shown that away from these singularities, a small perturbation from the integrable structure results in smooth near integrable behavior. Such a perturbation may occur from a small deformation or tilt of the wall which breaks the separability upon impact, the addition of a small regular perturbation to the system, or the combination of both. In some simple cases explicit formulae to the leading order term in the near integrable return map are derived. Near integrability is also shown to persist when the hard billiard boundary is replaced by a singular, smooth, steep potential, thus extending the near-integrability results beyond the scope of regular perturbations. These systems constitute an additional class of examples of near integrable impact systems, beyond the traditional one dimensional oscillating billiards, nearly elliptic billiards, and the near-integrable behavior near the boundary of convex smooth billiards with or without magnetic field.