2008/08/23 by Michael Bialy, Bialy, Michael
Mathematics · Physics and Astronomy · #37J50 #70H15 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #math.DG #math.DS #msc:37J50 #msc:70H15
paper · pdf · doi:10.48550/arxiv.0808.3208
arxiv created 2008/08/23 · openalex publication_date 2008/08/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The main result of this paper is, that for convex billiards in higher dimensions, in contrast with 2D case, for every point on the boundary and for every n there always exist billiard trajectories developing conjugate points at the n-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the Length functional. It turns out that this formula behaves differently with respect to "longitudinal" and "transversal" variations.