On the structure of Hamiltonian impact systems

Abstract

Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an additional rich class of non-smooth systems that can be analyzed. Such systems exhibit rich dynamics, as, in addition to the underlying integrable structure, some of the trajectories may undergo only transverse impact, others may undergo also tangential impacts, and some trajectories do not impact at all. Under perturbations, each of these classes of orbits behaves differently. Tools for classifying these different types of dynamics in 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable dynamics are presented. Moreover, some of these tools may be extended to far from integrable cases. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to impact systems is constructed. It is shown that such representations classify dynamically different regions in phase space. For the integrable and near integrable (small perturbations) cases these provide global information on the dynamics whereas for the far from integrable (non-perturbative) regimes, these provide rough classification of the first impact map. The interpretation of these results in terms of projections of solutions to the configuration space as well as the relations of these to the Hill region are presented.

Impact Hamiltonian systems and polygonal billiards

“…with θī defined by (20). This step-dynamics are also conjugated to the (±ω i (e i ), ±ωī(eī)) directional billiard motion on the rectangular billiard…”

“…Thus, by constructing a nibbled scattering geometry which combines finite and semi-infinite horizontal and vertical segments in the configuration space, the number of holes and the number of connected components in the iso-energy level set surfaces can be manipulated. Moreover, constructing an impact energy-momentum diagram [19,20], such as Fig 4 for the one-step system, allows to identify the critical energy values at which the topology of the energy surface changes.…”