Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

Abstract

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars

On the structure of Hamiltonian impact systems

“…Since each PRL is a rectangle which includes in its interior a point which is a local minimizer of V(by lemma 4.6), and since for exterior GWS all local minimizers are not in the billiard domain, we conclude that each PRL which includes a point inside the billiard domain must also include a minimizer of V which is not in the billiard domain, so it must intersect the wall, namely the leaf is an impact leaf as claimed. Since, by lemma 4.4 and equation (17), for H < H tmin min all the iso-energy leaves do not intersect the wall, by the above argument they all must lie outside of the billiard domain so no motion is allowed for H < H tmin min . Lemma 4.8 (Claim 2 of proposition 4.1).…”

“…Proof. Since, by lemma 4.4 and equation (17), for H < H tmin min all the iso-energy leaves do not intersect the wall, the leaves in the allowed region of motion belong to the non-impact zone. We need to check when some of these non-impacting leaves are in the allowed region of motion.…”

“…For energies in this wedge, the motion on each level set is conjugated to the directional motion on an L-shaped billiard with changing dimensions and direction, or, equivalently, to the directional motion on a flat surface of genus 2 [4]. The resulting IFG and their classification for the various cases are yet to be developed (the graphs of topological billiards appear to be relevant [17,18,29]).…”

“…H tmin (q 2 ) = H tmin (q w−min 2), (17) and the wall is tangent to the Hill region of H tmin min at q w (q w−min…”

“…Other recent works have presented Fomenko graphs and classification theory to integrable billiards in non-convex domains [29] and the more general topological billiards (see e.g. [17,18] and references within). For the billiards, the dynamics are independent of the energy and thus for any given table there is a single graph which represents how the dynamics depend on the constant of motion which is preserved by the integrable or quasi-integrable billiard motion.…”