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 | Abstract: | 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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