Email this article Level surfaces of the first integral for a billiard system with cosine refraction
- Authors: Nikulin M.A.1,2, Popelenskii F.Y.1,2,3
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Affiliations:
- Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
- Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
- National Research University Higher School of Economics, Moscow, Russia
- Issue: Vol 216, No 10 (2025)
- Pages: 101-158
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/331248
- DOI: https://doi.org/10.4213/sm10245
- ID: 331248
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Abstract
A new integrable system in an ellipse is introduced: the domain bounded by an 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. In crossing an interface between media the trajectory 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 arising when going over critical values of the 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 arising when going over critical values of the integral.
About the authors
Mikhail Aleksandrovich Nikulin
Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia; Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
Email: nikmihale@gmail.com
Fedor Yur'evich Popelenskii
Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia; Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia; National Research University Higher School of Economics, Moscow, Russia
Email: popelens@mech.math.msu.su
Candidate of physico-mathematical sciences, Associate professor
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