Impact Hamiltonian systems and polygonal billiards

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 two-dimensional surface in the four-dimensional phase space. Yet, for a range of energies, these surfaces are not the simple Liouville-Arnold tori - these are compact orientable surfaces 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 Liouville-Arnold 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 L-shaped billiards, where the billiard area and shape as well as the direction of motion vary continuously on isoenergetic level sets.