Abstract:
The authors have recently introduced the class of topological billiards. Topological billiards are glued from
elementary planar billiard sheets (bounded by arcs of confocal quadrics) along intervals of their boundaries. It turns out that the integrability of the elementary billiards implies that of the topological billiards. We show that all classical
linearly and quadratically integrable geodesic flows on tori and spheres are Liouville equivalent to appropriate topological billiards. Moreover, the linear and quadratic integrals of the geodesic flows reduce to a single
canonical linear integral and a single canonical quadratic integral on the billiard. These results are obtained within the
framework of the Fomenko–Zieschang theory of the classification of integrable systems.
This paper was written with the support of the Russian Federation President's Programme for
the support of leading scientific schools (grant no. NSh-6399.2018.1, contract no. 075-02-2018-867), and the Russian
Foundation for Basic Research (grant no. 19-01-00775-a).
Citation:
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
This publication is cited in the following 27 articles:
Anatoly Fomenko, “Hidden symmetries in Hamiltonian geometry, topology, physics and mechanics”, Priroda, 2025, no. 1(1313), 23
G. V. Belozerov, A. T. Fomenko, “Orbital invariants of billiards and linearly integrable geodesic flows”, Sb. Math., 215:5 (2024), 573–611
G. V. Belozerov, A. T. Fomenko, “Rotation Functions of Integrable Billiards As Orbital Invariants”, Dokl. Math., 2024
G. V. Belozerov, A. T. Fomenko, “Rotation functions of integrable billiards as orbital invariants”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 515:1 (2024), 5
JOSCHA HENHEIK, “Deformational rigidity of integrable metrics on the torus”, Ergod. Th. Dynam. Sys., 2024, 1
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. V. Vedyushkina, S. E. Pustovoitov, “Classification of Liouville foliations of integrable topological billiards in magnetic fields”, Sb. Math., 214:2 (2023), 166–196
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
M. A. Nikulin, “Spectrum of the Schrödinger operator in an elliptical ring cover”, Moscow University Mathematics Bulletin, 78:5 (2023), 230–243
Vladimir Dragović, Sean Gasiorek, Milena Radnović, “Billiard Ordered Games and Books”, Regul. Chaotic Dyn., 27:2 (2022), 132–150
A. T. Fomenko, V. V. Vedyushkina, “Evolutionary force billiards”, Izv. Math., 86:5 (2022), 943–979
V. V. Vedyushkina, V. N. Zav'yalov, “Realization of geodesic flows with a linear first integral by billiards with slipping”, Sb. Math., 213:12 (2022), 1645–1664
G. V. Belozerov, “Topology of 55-surfaces of a 3D billiard inside a triaxial ellipsoid with Hooke's potential”, Moscow University Mathematics Bulletin, 77:6 (2022), 277–289
A. T. Fomenko, V. A. Kibkalo, “Topology of Liouville foliations of integrable billiards on table-complexes”, European Journal of Mathematics, 8:4 (2022), 1392–1423
S. E. Pustovoitov, “Topological analysis of a billiard bounded by confocal quadrics in a potential field”, Sb. Math., 212:2 (2021), 211–233
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
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
S. E. Pustovoitov, “Topological analysis of an elliptic billiard in a fourth-order potential field”, Moscow University Mathematics Bulletin, 76:5 (2021), 193–205
A. T. Fomenko, V. V. Vedyushkina, V. N. Zav'yalov, “Liouville foliations of topological billiards with slipping”, Russ. J. Math. Phys., 28:1 (2021), 37–55
Citation in format AMSBIB
Contact us: