Liouville Integrable Generalized Billiard Flows and Poncelet Type Theorems

Abstract

“Glued geodesic flows” and, in particular, “generalized billiard flows” on Riemannian manifolds with boundary, and geodesic flows on piecewise smooth Riemannian manifolds are studied. We develop the approaches of Lazutkin (1993) and Tabachnikov (1993) for proving the Poncelet type closure theorems via applying the classical Liouville theorem to the billiard flow (respectively to the billiard map). We prove that the condition on the refraction/reflection law to respect the Huygens principle is not only sufficient, but also necessary for “local Liouville integrability” of the glued geodesic flow, more precisely for pairwise commutation of the “glued flows” corresponding to a maximal collection of local first integrals in involution homogeneous in momenta. A similar criterion for “local Liouville integrability” of the succession/billiard map is obtained.