2020/08/31 by Sean Gasiorek, Milena Radnović, Milena Radnovic
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Bounded function #Conic section #Dynamical billiards #Ellipse #Euclidean geometry #Generalization #Geometric Analysis and Curvature Flows #Geometry #Hyperboloid #Integrable system #Mathematical analysis #Mathematics #Mathematics and Applications #Minkowski space #Pure mathematics #Type (biology) #math.DS #msc:14H70 #msc:37J35 #msc:37J38 #msc:37J39 #msc:37J46 #msc:70H06 #msc:70H12
paper · pdf · doi:10.1016/j.geomphys.2020.104032
30 pages, 15 figures
openalex created_date 2020/08/21 · arxiv created 2020/10/27 · openalex publication_date 2020/11/27 · arxiv updated 2021/02/03 · openalex updated_date 2026/08/05
We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric integration technique, we prove that the billiard within generalized ellipses of each type is integrable in the sense of Liouville. Further, we prove a generalization of the Poncelet theorem and derive Cayley-type conditions for periodic trajectories and explore geometric consequences.