Authors: Dragović, Vladimir 
Radnović, Milena
Affiliations: Mechanics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Resonance of ellipsoidal billiard trajectories and extremal rational functions
Journal: Advances in Mathematics
Volume: 424
First page: 109044
Issue Date: 2023
Rank: ~M21
ISSN: 0001-8708
DOI: 10.1016/j.aim.2023.109044
We study resonant billiard trajectories within quadrics in the d-dimensional Euclidean space. We relate them to the theory of approximation, in particular the extremal rational functions on the systems of d intervals on the real line. This fruitful link enables us to prove fundamental properties of the billiard dynamics and to provide a comprehensive study of a large class of non-periodic trajectories of integrable billiards. A key ingredient is a functional-polynomial relation of a generalized Pell type. Applying further these ideas and techniques to s-weak billiard trajectories, we come to a functional-polynomial relation of the same generalized Pell type.
Keywords: Caustics | Cayley-type conditions | Ellipsoidal billiards | Elliptic and hyper-elliptic curves | Generalized Pell's equations | Resonant trajectories
Publisher: Elsevier

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