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Uspekhi Mat. Nauk, 2010, Volume 65, Issue 2(392), Pages 133–194 (Mi umn9349)  

This article is cited in 10 scientific papers (total in 10 papers)

Integrable billiards and quadrics

V. Dragovićab, M. Radnovića

a 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.

DOI: https://doi.org/10.4213/rm9349

Full text: PDF file (1258 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.938+531.01
MSC: Primary 37J35, 70J45; Secondary 58E07, 70H06
Received: 03.02.2010

Citation: V. Dragović, M. Radnović, “Integrable billiards and quadrics”, Uspekhi Mat. Nauk, 65:2(392) (2010), 133–194; Russian Math. Surveys, 65:2 (2010), 319–379

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    2. Jovanovic B., “Noncommutative Integrability and Action Angle Variables in Contact Geometry”, J. Symplectic Geom., 10:4 (2012), 535–561  crossref  mathscinet  zmath  isi  elib  scopus
    3. B. Jovanović, “The Jacobi-Rosochatius Problem on an Ellipsoid: the Lax Representations and Billiards”, Arch. Rational Mech. Anal., 210:1 (2013), 101–131  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Dragovic V., Radnovic M., “Minkowski Plane, Confocal Conics, and Billiards”, Publ. Inst. Math.-Beograd, 94:108 (2013), 17–30  crossref  mathscinet  zmath  isi  scopus
    5. Dragovic V., Radnovic M., “Bicentennial of the Great Poncelet Theorem (1813-2013): Current Advances”, Bull. Amer. Math. Soc., 51:3 (2014), 373–445  crossref  mathscinet  zmath  isi  scopus
    6. V. I. Dragović, M. Radnović, “Pseudo-integrable billiards and double reflection nets”, Russian Math. Surveys, 70:1 (2015), 1–31  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. V. Dragović, M. Radnović, “Topological invariants for elliptical billiards and geodesics on ellipsoids in the Minkowski space”, J. Math. Sci., 223:6 (2017), 686–694  mathnet  crossref  mathscinet  elib
    8. V. V. Kozlov, “Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas”, Russian Math. Surveys, 71:2 (2016), 253–290  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Kroetz T., Oliveira H.A., Portela J.S.E., Viana R.L., “Dynamical properties of the soft-wall elliptical billiard”, Phys. Rev. E, 94:2 (2016), 022218  crossref  mathscinet  isi  elib  scopus
    10. Schastnyy V. Treschev D., “on Local Integrability in Billiard Dynamics”, Exp. Math., 28:3 (2019), 362–368  crossref  isi
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