On some Impact-like Hamiltonian systems

The dynamics associated with mechanical Hamiltonian systems with smooth potentials that include sharp fronts is traditionally modeled by Hamiltonian impact systems:  a class of generalized billiards by which the dynamics in the domain’s interior are governed by smooth potentials and at the domain’s boundaries by elastic reflections. 

I will first discuss the properties of this singular limit, culminating in the recent work with D. Turaev in which we established the non-ergodicity of smooth N repelling particles in a box at arbitrarily high energy (in contrast to the common ergodic hypothesis). 

Then, I will introduce the class of quasi-integrable Hamiltonian impact systems, showing that the motion on some level sets is conjugated to a directed motion on a translation surface of a genus larger than one. We propose mechanical realizations of such systems, analyze ergodic properties and quantum properties of classes of such systems, and study their behavior under perturbations, leading to the study of piecewise smooth symplectic maps  (in joint works with L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, I. Pazi, M. Pnueli, K. Fraczek, O. Yaniv and A. Zobova)