Impact Hamiltonian systems and polygonal billiards
Abstract
The dynamics of a beam held on a horizontal frame by springs and bouncing off a step is described by a separable two degrees of freedom Hamiltonian system with impacts that respect, point wise, the separability symmetry. The energy in each degree of freedom is preserved, and the motion along each level set is conjugated, via action angle coordinates, to a geodesic flow on a flat twodimensional surface in the four dimensional phase space. Yet, for a range of energies, these surfaces are not the simple LiouvilleArnold tori  these are tori of genus two, thus the motion on them is not conjugated to simple rotations. Namely, even though energy is not transferred between the two degrees of freedom, the impact system is quasiintegrable and is not of the LiouvilleArnold type. In fact, for each level set in this range, the motion is conjugated to the well studied and highly nontrivial dynamics of directional motion in Lshaped billiards, where the billiard area and shape as well as the direction of motion vary continuously on isoenergetic level sets. Return maps to Poincaré section of the flow are shown to be conjugated, on each level set, to interval exchange maps which are computed, up to quadratures, in the general nonlinear case and explicitly for the case of two linear oscillators bouncing off a step. It is established that for any such oscillatorstep system there exist step locations for which some of the level sets exhibit motion which is neither periodic nor ergodic. Changing the impact surface by introducing additional steps, staircases, strips and blocks from which the particle is reflected, leads to isoenergy surfaces that are foliated by families of genusk level set surfaces, where the number and order of families of genus k depend on the energy.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 DOI:
 10.48550/arXiv.2001.03726
 arXiv:
 arXiv:2001.03726
 Bibcode:
 2020arXiv200103726B
 Keywords:

 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Chaotic Dynamics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 To appear in the "Proceedings of the MSRI 2018 Fall semester on Hamiltonian Systems"