Abstract:
We consider a generalization of a mathematical billiard bounded by arcs of confocal quadrics, known as billiard books. Billiard books define a large class of integrable Hamiltonian systems. In this connection the question arises of the possibility of realizing integrable Hamiltonian systems with two degrees of freedom by billiard books. The authors have proved previously that for any nondegenerate three-dimensional bifurcation (33-atom) a billiard book in which such a bifurcation appears can be constructed algorithmically. Based on the preceding result, we give a proof of the fact that given any base of a Liouville foliation (rough molecule), a billiard book can be constructed algorithmically such that the base of the Liouville foliation of this system is isomorphic to the one given initially.
Bibliography: 15 titles.
The research of I. S. Kharcheva was supported by the Russian Foundation for Basic Research (grant no. 19-01-00775-a) and by the Theoretical Physics and Mathematics Advancement Foundation “BASIS” (grant no. 19-8-2-5-1).
Citation:
V. V. Vedyushkina, I. S. Kharcheva, “Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems”, Sb. Math., 212:8 (2021), 1122–1179
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\paper Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems
\jour Sb. Math.
\yr 2021
\vol 212
\issue 8
\pages 1122--1179
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Linking options:
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This publication is cited in the following 14 articles:
Anatoly Fomenko, “Hidden symmetries in Hamiltonian geometry, topology, physics and mechanics”, Priroda, 2025, no. 1(1313), 23
K. E. Turina, “Topological invariants of some ordered billiard games”, Moscow University Mathematics Bulletin, 79:3 (2024), 122–129
D. A. Tuniyants, “Topology of isoenergetic surfaces of billiard books glued of rings”, Moscow University Mathematics Bulletin, 79:3 (2024), 130–141
V. A. Kibkalo, D. A. Tuniyants, “Uporyadochennye billiardnye igry i topologicheskie svoistva billiardnykh knizhek”, Trudy Voronezhskoi zimnei matematicheskoi shkoly S. G. Kreina — 2024, SMFN, 70, no. 4, Rossiiskii universitet druzhby narodov, M., 2024, 610–625
V. N. Zav'yalov, “Billiard with slipping by an arbitrary rational angle”, Sb. Math., 214:9 (2023), 1191–1211
A. T. Fomenko, V. V. Vedyushkina, “Billiards and integrable systems”, Russian Math. Surveys, 78:5 (2023), 881–954
A. A. Kuznetsova, “Modeling of degenerate peculiarities of integrable billiard systems by billiard books”, Moscow University Mathematics Bulletin, 78:5 (2023), 207–215
G. V. Belozerov, “Topological classification of billiards bounded by confocal quadrics in three-dimensional Euclidean space”, Sb. Math., 213:2 (2022), 129–160
A. T. Fomenko, V. V. Vedyushkina, “Evolutionary force billiards”, Izv. Math., 86:5 (2022), 943–979
A. T. Fomenko, V. A. Kibkalo, “Topology of Liouville foliations of integrable billiards on table-complexes”, Eur. J. Math., 8:4 (2022), 1392–1423
V. V. Vedyushkina, V. A. Kibkalo, “Billiardnye knizhki maloi slozhnosti i realizatsiya sloenii Liuvillya integriruemykh sistem”, Chebyshevskii sb., 23:1 (2022), 53–82
A. T. Fomenko, V. V. Vedyushkina, “Billiards with changing geometry and their connection with the implementation of the Zhukovsky and Kovalevskaya cases”, Russ. J. Math. Phys., 28:3 (2021), 317–332
V. Kibkalo, A. Fomenko, I. Kharcheva, “Realizing integrable Hamiltonian systems by means of billiard books”, Trans. Moscow Math. Soc., 82 (2021), 37–64
V. V. Vedyushkina, V. A. Kibkalo, S. E. Pustovoitov, “Realizatsiya fokusnykh osobennostei integriruemykh sistem billiardnymi knizhkami s potentsialom Guka”, Chebyshevskii sb., 22:5 (2021), 44–57
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