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Integrable billiards and quadrics. (English. Russian original) Zbl 1202.37078
Russ. Math. Surv. 65, No. 2, 319-379 (2010); translation from Usp. Mat. Nauk 65, No. 2, 133-194 (2010).
The main aim of this impressive paper is to give a detailed survey of the last twenty years concerning dynamical and analytical properties of billiard systems, with a special view towards integrability. Starting with billiards inside conic sections, the authors consider billiards inside quadratics as their natural higher-dimensional generalizations. Also, the interplay between billiard dynamics, linear subspaces of intersections of quadrics, and hyperelliptic Jacobians enabled the authors to obtain higher-dimensional generalizations of several classical results as the Poncelet theorem (1813, 1822), the Darboux theorem (1870), and the Weyr theorem (1870).

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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