This article is cited in 22 papers

Integrable billiards and quadrics

V. Dragović^ab, M. Radnović<sup>a

a</sup> Mathematical Institute SASA, Belgrade, Serbia
^b Mathematical Physics Group, University of Lisbon, Portugal

Abstract: Billiards inside quadrics are considered as integrable dynamical systems with a rich geometric structure. The two-way interaction between the dynamics of billiards and the geometry of pencils of quadrics in an arbitrary dimension is considered. Several well-known classical and modern genus-1 results are generalized to arbitrary dimension and genus, such as: the Poncelet theorem, the Darboux theorem, the Weyr theorem, and the Griffiths–Harris space theorem. A synthetic approach to higher-genera addition theorems is presented.
Bibliography: 77 titles.

Keywords: hyperelliptic curve, Jacobian variety, Poncelet porism, periodic trajectories, Poncelet–Darboux grids, addition theorems.

UDC: 517.938+531.01

MSC: Primary 37J35, 70J45; Secondary 58E07, 70H06

Received: 03.02.2010

DOI: 10.4213/rm9349

 English version: Russian Mathematical Surveys, 2010, 65:2, 319–379

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