Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems

V. V. Vedyushkina   ; I. S. Kharcheva

Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1690-1727

Résumé

We introduce a new class of billiards—billiard books, which are integrable Hamiltonian systems. It turns out that for any nondegenerate three-dimensional bifurcation (3-atom), a billiard book can be algorithmically constructed in which such a bifurcation appears. Consequently, any integrable Hamiltonian nondegenerate dynamical system with two degrees of freedom can be modelled in some neighbourhood of a critical leaf of the Liouville foliation in the iso-energy 3-manifold by a billiard. Bibliography: 25 titles.

Détail

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

Bibliographie

[1] G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York, 1927, viii+295 pp. | MR | Zbl | Zbl

[2] 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. | MR | MR | Zbl | Zbl

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

[4] V. Dragović, M. Radnović, Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics, Front. Math., Birkhäuser/Springer Basel AG, Basel, 2011, viii+293 pp. | DOI | MR | Zbl

[5] V. V. Fokicheva, “Description of singularities for system “billiard in an ellipse” ”, Moscow Univ. Math. Bull., 67:5-6 (2012), 217–220 | DOI | MR | Zbl

[6] 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

[7] 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

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

[10] 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

[11] E. A. Kudryavtseva, I. M. Nikonov, A. T. Fomenko, “Maximally symmetric cell decompositions of surfaces and their coverings”, Sb. Math., 199:9 (2008), 1263–1353 | DOI | DOI | MR | Zbl

[12] E. A. Kudryavtseva, A. T. Fomenko, “Symmetries groups of nice Morse functions on surfaces”, Dokl. Math., 86:2 (2012), 691–693 | DOI | MR | Zbl

[13] A. V. Bolsinov, S. V. Matveev, A. T. Fomenko, “Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of systems of small complexity”, Russian Math. Surveys, 45:2 (1990), 59–94 | DOI | MR | Zbl

[14] A. T. Fomenko, Symplectic geometry, Adv. Stud. Contemp. Math., 5, 2nd ed., Gordon and Breach Publishers, Luxembourg, 1995, xvi+467 pp. | MR | MR | Zbl | Zbl

[15] A. T. Fomenko, “Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body”, Differentsialnaya geometriya, gruppy Li i mekhanika. 15–2, Zap. nauch. sem. POMI, 235, POMI, SPb., 1996, 104–183 ; J. Math. Sci. (N. Y.), 94:4 (1999), 1512–1557 | MR | Zbl | DOI

[16] A. V. Bolsinov, P. H. Richter, A. T. Fomenko, “The method of loop molecules and the topology of the Kovalevskaya top”, Sb. Math., 191:2 (2000), 151–188 | DOI | DOI | MR | Zbl

[17] A. T. Fomenko, A. Yu. Konyaev, “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topology Appl., 159:7 (2012), 1964–1975 | DOI | MR | Zbl

[18] A. A. Oshemkov, “Classification of hyperbolic singularities of rank zero of integrable Hamiltonian systems”, Sb. Math., 201:8 (2010), 1153–1191 | DOI | DOI | MR | Zbl

[19] A. A. Oshemkov, “The topology of the set of singularities for an integrable Hamiltonian system”, Dokl. Math., 82:2 (2010), 777–779 | DOI | MR | Zbl

[20] A. T. Fomenko, A. Konyaev, “Algebra and geometry through Hamiltonian systems”, Continuous and distributed systems. Theory and applications, Solid Mech. Appl., 211, Springer, Cham, 2014, 3–21 | DOI | MR | Zbl

[21] A. T. Fomenko, S. S. Nikolaenko, “The Chaplygin case in dynamics of a rigid body in fluid is orbitally equivalent to the Euler case in rigid body dynamics and to the Jacobi problem about geodesics on the ellipsoid”, J. Geom. Phys., 87 (2015), 115–133 | DOI | MR | Zbl

[22] V. V. Fokicheva, “Classification of billiard motions in domains bounded by confocal parabolas”, Sb. Math., 205:8 (2014), 1201–1221 | DOI | DOI | MR | Zbl

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

[24] 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

[25] C. G. J. Jacobi, Vorlesungen über Dynamik, Gesammelte Werke, Supplementband, 2. rev. Ausg., G. Reimer, Berlin, 1884, viii+300 pp. | MR | Zbl