Tools for analyzing dynamics in a class of 2 degrees-of-freedom Hamiltonian impact systems with underlying separable integrable structure are derived. Integrable, near-integrable and far-from integrable cases are considered. In particular, a generalization of the energy momentum bifurcation diagram, Fomenko graphs and the hierarchy of bifurcations framework to this class is constructed. The projection of Liouville leaves of the smooth integrable dynamics to the configuration space allows to extend these tools to impact surfaces which produce far from integrable dynamics. It is suggested that such representations classify dynamically different regions in phase space. For the integrable and near integrable cases these provide global information on the dynamics whereas for the far from integrable regimes (caused by finite deformations of the impact surface), these provide information on the singular set and on the non-impact orbits. The results are presented and demonstrated for the Duffing-center system with impacts from a slanted wall.

导出了用于分析具有潜在可分离可积结构的一类 2 自由度哈密顿碰撞系统中的动力学的工具。考虑可积、近可积和远不可积的情况。特别是，构建了对此类的能量动量分叉图、Fomenko 图和分叉框架的层次结构的推广。平滑可积动力学的刘维尔叶投影到配置空间允许将这些工具扩展到产生远离可积动力学的冲击表面。建议这种表示对相空间中的不同区域进行动态分类。对于可积和近可积的情况，这些提供了动力学的全局信息，而对于远不可积的情况（由撞击表面的有限变形引起），这些提供了关于奇异集和非撞击轨道的信息。对 Duffing-center 系统的结果进行了展示和演示，该系统受到倾斜墙的影响。