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Fundam. Prikl. Mat., 2015, Volume 20, Issue 3, Pages 113–152 (Mi fpm1656)  

This article is cited in 2 scientific papers (total in 2 papers)

Liouville integrable generalized billiard flows and Poncelet type theorems

E. A. Kudryavtseva

Lomonosov Moscow State University

Abstract: “Glued geodesic flows” and, in particular, “generalized billiard flows” on Riemannian manifolds with boundary, and geodesic flows on piecewise smooth Riemannian manifolds are studied. We develop the approaches of Lazutkin (1993) and Tabachnikov (1993) for proving the Poncelet type closure theorems via applying the classical Liouville theorem to the billiard flow (respectively to the billiard map). We prove that the condition on the refraction/reflection law to respect the Huygens principle is not only sufficient, but also necessary for “local Liouville integrability” of the glued geodesic flow, more precisely for pairwise commutation of the “glued flows” corresponding to a maximal collection of local first integrals in involution homogeneous in momenta. A similar criterion for “local Liouville integrability” of the succession/billiard map is obtained.

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English version:
Journal of Mathematical Sciences (New York), 2017, 225:4, 611–638

Bibliographic databases:

UDC: 514.853+517.938.5

Citation: E. A. Kudryavtseva, “Liouville integrable generalized billiard flows and Poncelet type theorems”, Fundam. Prikl. Mat., 20:3 (2015), 113–152; J. Math. Sci., 225:4 (2017), 611–638

Citation in format AMSBIB
\Bibitem{Kud15}
\by E.~A.~Kudryavtseva
\paper Liouville integrable generalized billiard flows and Poncelet type theorems
\jour Fundam. Prikl. Mat.
\yr 2015
\vol 20
\issue 3
\pages 113--152
\mathnet{http://mi.mathnet.ru/fpm1656}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3519751}
\elib{http://elibrary.ru/item.asp?id=31050064}
\transl
\jour J. Math. Sci.
\yr 2017
\vol 225
\issue 4
\pages 611--638
\crossref{https://doi.org/10.1007/s10958-017-3482-5}


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    This publication is cited in the following articles:
    1. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173  mathnet  crossref  crossref  adsnasa  isi
  • Фундаментальная и прикладная математика
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