Billiard with slipping by an arbitrary rational angle

V. N. Zav'yalov

Sbornik. Mathematics, Tome 214 (2023) no. 9, pp. 1191-1211

Résumé

The class of billiards in a disc with slipping along the boundary circle by an angle commensurable with $\pi$ is considered. For such billiards it is shown that an isoenergy surface of the system is homeomorphic to a lens space $L(q,p)$ with parameters satisfying $0 < p <q$. The set of pairs $(q, p)$ such that there exists a billiard in a disc realizing the corresponding lens space $L(q,p)$ is described in terms of solutions of a linear Diophantine equation in two variables. This result also holds for planar billiards with slipping in simply connected domains with smooth boundary, that is, it is not confined to the integrable case. Bibliography: 30 titles.

Détail

Keywords: integrable system, slipping
Mots-clés : billiard, Fomenko-Zieschang invariant, lens space.

Bibliographie

[1] A. A. Glutsyuk, “On polynomially integrable Birkhoff billiards on surfaces of constant curvature”, J. Eur. Math. Soc. (JEMS), 23:3 (2021), 995–1049 | DOI | MR | Zbl

[2] M. Bialy and A. E. Mironov, “Algebraic non-integrability of magnetic billiards”, J. Phys. A, 49:45 (2016), 455101, 18 pp. | DOI | MR | Zbl

[3] V. Kaloshin and A. Sorrentino, “On the local Birkhoff conjecture for convex billiards”, Ann. of Math. (2), 188:1 (2018), 315–380 | DOI | MR | Zbl

[4] V. V. Vedyushkina and I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems”, Sb. Math., 209:12 (2018), 1690–1727 | DOI | MR | Zbl

[5] V. V. Vedyushkina, A. T. Fomenko and I. S. Kharcheva, “Modeling nondegenerate bifurcations of closures of solutions for integrable systems with two degrees of freedom by integrable topological billiards”, Dokl. Math., 97:2 (2018), 174–176 | DOI | MR | Zbl

[6] V. V. Fokicheva, “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics”, Sb. Math., 206:10 (2015), 1463–1507 | DOI | MR | Zbl

[7] V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable topological billiards and equivalent dynamical systems”, Izv. Math., 81:4 (2017), 688–733 | DOI | MR | Zbl

[8] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | DOI | MR | Zbl

[9] A. T. Fomenko, “Morse theory of integrable Hamiltonian systems”, Soviet Math. Dokl., 33:2 (1986), 502–506 | MR | Zbl

[10] A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658 | DOI | MR | Zbl

[11] A. T. Fomenko and H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom”, Math. USSR-Izv., 36:3 (1991), 567–596 | DOI | MR | Zbl

[12] V. Dragović and M. Radnović, “Bifurcations of Liouville tori in elliptical billiards”, Regul. Chaotic Dyn., 14:4–5 (2009), 479–494 | DOI | MR | Zbl

[13] V. V. Fokicheva (Vedyushkina), “Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas”, Moscow Univ. Math. Bull., 69:4 (2014), 148–158 | DOI | MR | Zbl

[14] V. V. Vedyushkina, “The Fomenko-Zieschang invariants of nonconvex topological billiards”, Sb. Math., 210:3 (2019), 310–363 | DOI | MR | Zbl

[15] V. V. Fokicheva (Vedyushkina) and A. T. Fomenko, “Integrable billiards model important integrable cases of rigid body dynamics”, Dokl. Math., 92:3 (2015), 682–684 | DOI | MR | Zbl

[16] V. V. Vedyushkina, “The Liouville foliation of the billiard book modelling the Goryachev-Chaplygin case”, Moscow Univ. Math. Bull., 75:1 (2020), 42–46 | DOI | MR | Zbl

[17] V. V. Vedyushkina and A. T. Fomenko, “Reducing the degree of integrals of Hamiltonian systems by using billiards”, Dokl. Math., 99:3 (2019), 266–269 | DOI | MR | Zbl

[18] V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards”, Izv. Math., 83:6 (2019), 1137–1173 | DOI | MR | Zbl

[19] A. T. Fomenko and V. V. Vedyushkina, “Billiards and integrability in geometry and physics. New scope and new potential”, Moscow Univ. Math. Bull., 74:3 (2019), 98–107 | DOI | MR | Zbl

[20] V. V. Vedyushkina and I. S. Kharcheva, “Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems”, Sb. Math., 212:8 (2021), 1122–1179 | DOI | MR | Zbl

[21] V. V. Vedyushkina, V. A. Kibkalo and A. T. Fomenko, “Topological modeling of integrable systems by billiards: realization of numerical invariants”, Dokl. Math., 102:1 (2020), 269–271 | DOI | MR | Zbl

[22] V. V. Vedyushkina, “Local modeling of Liouville foliations by billiards: implementation of edge invariants”, Moscow Univ. Math. Bull., 76:2 (2021), 60–64 | DOI | MR | Zbl

[23] V. V. Vedyushkina and V. A. Kibkalo, “Realization of the numerical invariant of the Seifert fibration of integrable systems by billiards”, Moscow Univ. Math. Bull., 75:4 (2020), 161–168 | DOI | MR | Zbl

[24] V. V. Vedyushkina and V. A. Kibkalo, “Billiard books of small complexity and realization of Liouville foliations for integrable systems”, Chebyshevskii Sb., 23:1 (2022), 53–82 (Russian) | DOI | MR

[25] I. S. Kharcheva, “Isoenergetic manifolds of integrable billiard books”, Moscow Univ. Math. Bull., 75:4 (2020), 149–160 | DOI | MR | Zbl

[26] V. V. Vedyushkina, “Integrable billiard systems realize toric foliations on lens spaces and the 3-torus”, Sb. Math., 211:2 (2020), 201–225 | DOI | MR | Zbl

[27] V. V. Vedyushkina, “Topological type of isoenergy surfaces of billiard books”, Sb. Math., 212:12 (2021), 1660–1674 | DOI | MR | Zbl

[28] A. T. Fomenko, V. V. Vedyushkina and V. N. Zav'yalov, “Liouville foliations of topological billiards with slipping”, Russ. J. Math. Phys., 28:1 (2021), 37–55 | DOI | MR | Zbl

[29] V. V. Kozlov, “Topological obstructions to the integrability of natural mechanical systems”, Soviet Math. Dokl., 20:6 (1979), 1413–1415 | MR | Zbl

[30] V. V. Vedyushkina and V. N. Zav'yalov, “Realization of geodesic flows with a linear first integral by billiards with slipping”, Sb. Math., 213:12 (2022), 1645–1664 | DOI | MR