RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mosc. Math. J.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mosc. Math. J., 2002, Volume 2, Number 1, Pages 183–196 (Mi mmj51)  

This article is cited in 8 scientific papers (total in 8 papers)

Ellipsoids, complete integrability and hyperbolic geometry

S. L. Tabachnikov

Pennsylvania State University

Abstract: We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperbolic metric inside the ellipsoid (the Caley–Klein model of hyperbolic space).

Key words and phrases: Riemannian and Finsler metrics, symplectic and contact structures, geodesic flow, mathematical billiard, hyperbolic metric, Caley–Klein model, exact transverse line fields.

DOI: https://doi.org/10.17323/1609-4514-2002-2-1-183-196

Full text: http://www.ams.org/.../abst2-1-2002.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 53A15, 53A20, 53D25
Received: October 30, 2001; in revised form January 15, 2002
Language:

Citation: S. L. Tabachnikov, “Ellipsoids, complete integrability and hyperbolic geometry”, Mosc. Math. J., 2:1 (2002), 183–196

Citation in format AMSBIB
\Bibitem{Tab02}
\by S.~L.~Tabachnikov
\paper Ellipsoids, complete integrability and hyperbolic geometry
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 1
\pages 183--196
\mathnet{http://mi.mathnet.ru/mmj51}
\crossref{https://doi.org/10.17323/1609-4514-2002-2-1-183-196}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1900590}
\zmath{https://zbmath.org/?q=an:1013.37029}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208587700010}
\elib{http://elibrary.ru/item.asp?id=8379101}


Linking options:
  • http://mi.mathnet.ru/eng/mmj51
  • http://mi.mathnet.ru/eng/mmj/v2/i1/p183

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Boualem H., Brouzet R., Rakotondralambo J., “About the separability of completely integrable quasi-bi-Hamiltonian systems with compact levels”, Differential Geometry and Its Applications, 26:6 (2008), 583–591  crossref  mathscinet  zmath  isi
    2. Albouy A., “Projective dynamics and classical gravitation”, Regular & Chaotic Dynamics, 13:6 (2008), 525–542  crossref  mathscinet  zmath  adsnasa  isi
    3. V. Dragović, M. Radnović, “Integrable billiards and quadrics”, Russian Math. Surveys, 65:2 (2010), 319–379  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Duval C., Valent G., “A new integrable system on the sphere and conformally equivariant quantization”, J Geom Phys, 61:8 (2011), 1329–1347  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Alain Albouy, “Projective Dynamics and First Integrals”, Regul. Chaotic Dyn., 20:3 (2015), 247–276  mathnet  crossref  mathscinet  zmath  adsnasa
    6. Albouy A., “on the Force Fields Which Are Homogeneous of Degree-3”, Extended Abstracts Spring 2014: Hamiltonian Systems and Celestial Mechanics; Virus Dynamics and Evolution, Trends in Mathematics, eds. Corbera M., Cors J., Llibre J., Birkhauser Boston, 2015, 3–7  crossref  isi
    7. Jovanovic B. Jovanovic V., “Virtual billiards in pseudo-Euclidean spaces: discrete Hamiltonian and contact integrability”, Discret. Contin. Dyn. Syst., 37:10 (2017), 5163–5190  crossref  zmath  isi  scopus
    8. Jovanovic B., “Billiards on Constant Curvature Spaces and Generating Functions For Systems With Constraints”, Theor. Appl. Mech., 44:1 (2017), 103–114  crossref  isi  scopus
  • Moscow Mathematical Journal
    Number of views:
    This page:192
    References:32

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019