Level surfaces of the first integral for a billiard system with cosine refraction

M. A. Nikulin^ab, F. Yu. Popelenskii<sup>abc

a</sup> Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
^b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^c National Research University Higher School of Economics, Moscow, Russia

Abstract: A new integrable system in an ellipse is introduced: the domain bounded by the ellipse is partitioned into subdomains by arcs of confocal quadrics, and each subdomain is filled by a medium with fixed constant coefficient of ‘optical’ density. The trajectory crossing an interface between media obeys the ‘cosine law’ of refraction. It is shown that such systems have an additional first integral.
For two partitions of an ellipse into subdomains the level surfaces of the additional integral are examined in detail, as well as their bifurcations occurring in going over critical values of the integral.
Bibliography: 21 titles.

Keywords: integrable billiard, confocal quadrics, law of refraction, bifurcations.

MSC: 37C83, 37J35

Received: 18.12.2024

DOI: 10.4213/sm10245

 English version: Sbornik: Mathematics, 2025, 216:10, 1428–1482

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