Another billiard problem
Sergey Bolotin, Dmitry Treschev
TL;DR
The paper develops a regularization framework for the geodesic flow of a degenerate metric $G=g/\phi$ on a domain $\Omega$ with boundary $\Gamma$, where $\phi>0$ in $\Omega$ and $\phi|_{\Gamma}=0$. By doubling the domain and applying Levi-Civita regularization, the incomplete geodesic flow is extended to a complete flow, yielding a billiard-like map on $T^*\Gamma$ without requiring convexity. Through semigeodesic coordinates and action-angle techniques, the authors derive a normal form for the Hamiltonian near the boundary and show that the billiard map is well-approximated by a time-$\pi$ map of an averaged Hamiltonian, enabling KAM-type results and Lazutkin-type theorems that guarantee the existence of invariant tori near the boundary. The work also identifies integrable special cases and discusses isoenergetic reductions, averaging, and shadowing of geodesic dynamics by the regularized billiard flow, highlighting the impact on short-wave approximations for degenerate wave equations. Overall, the paper provides a robust mechanism to study billiard-like dynamics in degenerate geometries and elucidates the persistence of quasi-integrable structures in this setting.
Abstract
Let $(M,g)$ be a Riemannian manifold, $Ω\subset M$ a domain with boundary $Γ$, and $φ$ a smooth function such that $φ|_Ω> 0$, $\ph|_Γ= 0$, and $\nablaφ|_Γ\ne 0$. We study the geodesic flow of the metric $G=g/φ$. The $G$-distance from any point of $Ω$ to $Γ$ is finite, hence the geodesic flow is incomplete. Regularization of the flow in a neighborhood of $Γ$ establishes a natural reflection law from $Γ$. This leads to a certain billiard problem in $Ω$.