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This article is cited in 49 scientific papers (total in 49 papers)
Integrable topological billiards and equivalent dynamical systems
V. V. Vedyushkina (Fokicheva), A. T. Fomenko Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We consider several topological integrable billiards and prove that they
are Liouville equivalent to many systems of rigid body dynamics. The proof
uses the Fomenko–Zieschang theory of invariants of integrable systems.
We study billiards bounded by arcs of confocal quadrics and their
generalizations, generalized billiards, where the motion occurs
on a locally planar surface obtained by gluing several planar domains
isometrically along their boundaries, which are arcs of confocal quadrics.
We describe two new classes of integrable billiards bounded by arcs
of confocal quadrics, namely, non-compact billiards and generalized billiards
obtained by gluing planar billiards along non-convex parts of their boundaries.
We completely classify non-compact billiards bounded by arcs of confocal
quadrics and study their topology using the Fomenko invariants that describe
the bifurcations of singular leaves of the additional integral. We study the
topology of isoenergy surfaces for some non-convex generalized billiards.
It turns out that they possess exotic Liouville foliations: the integral
trajectories of the billiard that lie on some singular leaves admit no
continuous extension. Such billiards appear to be leafwise equivalent
to billiards bounded by arcs of confocal quadrics in the Minkowski metric.
Keywords:
integrable system, billiard, Liouville equivalence, Fomenko–Zieschang molecule.
Received: 15.09.2016
Citation:
V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. RAN. Ser. Mat., 81:4 (2017), 20–67; Izv. Math., 81:4 (2017), 688–733
Linking options:
https://www.mathnet.ru/eng/im8602https://doi.org/10.1070/IM8602 https://www.mathnet.ru/eng/im/v81/i4/p20
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Abstract page: | 866 | Russian version PDF: | 172 | English version PDF: | 25 | References: | 72 | First page: | 52 |
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