# On the structure of Hamiltonian impact systems

## TL;DR

This paper develops tools to classify and analyze the complex dynamics of Hamiltonian impact systems, including impact types and bifurcations, extending classical integrable system frameworks to non-smooth, impact-affected systems.

## Contribution

It introduces a framework extending energy-momentum bifurcation diagrams and Fomenko graphs to impact systems, enabling classification of phase space regions in integrable and near-integrable Hamiltonian impacts.

## Key findings

- Classifies impact dynamics using generalized bifurcation diagrams.
- Provides tools for global phase space analysis in near-integrable impact systems.
- Extends classification methods to far-from-integrable regimes.

## 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.