Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems

V. V. Vedyushkina ; I. S. Kharcheva

Sbornik. Mathematics, Tome 212 (2021) no. 8, pp. 1122-1179

Résumé

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 ($3$-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.

Détail

Keywords: mathematical billiard, Hamiltonian system
Mots-clés : Liouville equivalence, Fomenko-Zieschang invariant.

Bibliographie

[1] V. V. Kozlov, D. V. Treshchev, Billiards. A genetic introduction to the dynamics of systems with impacts, Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991, viii+171 pp. | DOI | MR | MR | Zbl | Zbl

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

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

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

[5] 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 | DOI | DOI | MR | Zbl

[6] A. T. Fomenko, “The symplectic topology of completely integrable Hamiltonian systems”, Russian Math. Surveys, 44:1 (1989), 181–219 | DOI | MR | Zbl

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

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

[9] A. T. Fomenko, 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

[10] V. I. Dragović, M. Radnović, “Pseudo-integrable billiards and double reflection nets”, Russian Math. Surveys, 70:1 (2015), 1–31 | DOI | DOI | MR | Zbl

[11] V. A. Moskvin, “Topology of Liouville bundles of integrable billiards in non-convex domains”, Moscow Univ. Math. Bull., 73:3 (2018), 103–110 | DOI | MR | Zbl

[12] 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 | DOI | MR | Zbl

[13] A. T. Fomenko, V. V. Vedyushkina, “Topological billiards, conservation laws and classification of trajectories”, Functional analysis and geometry: Selim Grigorievich Krein centennial, Contemp. Math., 733, Amer. Math. Soc., Providence, RI, 2019, 129–148 | DOI | MR | Zbl

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

[15] V. V. Vedyushkina, A. T. Fomenko, 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 | DOI | MR | Zbl