Geodesic
Topological analysis of a~billiard in elliptic ring in a~potential field
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 6, pp. 201-225.

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We study a billiard in a domain bounded by two confocal ellipses. The Hooke potential is placed at the center of the ellipses. This dynamic system turns out to be Liouville integrable. Therefore, we can make a topological analysis studying the foliation of the phase manifold by integrals. We calculate Fomenko–Zieschang invariants (marked molecules) for isoenergy manifolds of every level of the Hamiltonian, and also give examples of other integrable systems that are Liouville equivalent to our billiard system. 
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S. E. Pustovoytov. Topological analysis of a~billiard in elliptic ring in a~potential field. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 6, pp. 201-225. http://geodesic.mathdoc.fr/item/FPM_2019_22_6_a8/

[1] Birkgof Dzh., Dinamicheskie sistemy, Gostekhizdat, M.–L., 1936

[2] Bolsinov A. V., Fomenko A. T., Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, v. 1, RKhD, Izhevsk, 1999

[3] Vedyushkina (Fokicheva) V. V., Fomenko A. T., “Integriruemye topologicheskie billiardy i ekvivalentnye dinamicheskie sistemy”, Izv. RAN. Ser. matem., 81:4 (2017), 20–67 | DOI | MR | Zbl

[4] Vedyushkina V. V., Fomenko A. T., Kharcheva I. S., “Modelirovanie nevyrozhdennykh bifurkatsii zamykanii reshenii integriruemykh sistem s dvumya stepenyami svobody integriruemymi topologicheskimi billiardami”, Dokl. RAN, 479:6 (2018), 607–610 | DOI | MR | Zbl

[5] Kobtsev I. F., Topologiya sloeniya Liuvillya integriruemogo billiarda v potentsialnom pole, Kurs. rabota, Mekhaniko-matematicheskii fakultet MGU. Kafedra differentsialnoi geometrii i prilozhenii, 2016

[6] Kozlov V. V., “Nekotorye integriruemye obobscheniya zadachi Yakobi o geodezicheskikh na ellipsoide”, Prikl. matem. i mekh., 59:1 (1995), 3–9 | MR | Zbl

[7] Fedoseev D. A., Fomenko A. T., “Nekompaktnye osobennosti integriruemykh dinamicheskikh sistem”, Fundam. i prikl. matem., 21:6 (2016), 217–243 | MR

[8] Fokicheva V. V., “Topologicheskaya klassifikatsiya billiardov v lokalno ploskikh oblastyakh, ogranichennykh dugami sofokusnykh kvadrik”, Matem. sb., 206:10 (2015), 127–176 | DOI | MR | Zbl

[9] Fokicheva V. V., Fomenko A. T., “Integriruemye billiardy modeliruyut vazhnye integriruemye sluchai dinamiki tverdogo tela”, Dokl. RAN, 465:2 (2015), 150–153 | DOI | MR | Zbl

[10] Kharlamov M. P., “Topologicheskii analiz i bulevy funktsii. I. Metody i priblizheniya k klassicheskim sistemam”, Nelineinaya dinamika, 6:4 (2010), 769–805

[11] Yakobi K., Lektsii po dinamike, Gostekhizdat, M., 1936

[12] Fokicheva V. V., Fomenko A. T., “Billiard systems as the models for the rigid body dynamics”, Studies in Systems, Decision and Control, Adv. Dynam. Syst. Control, 69, eds. V. A. Sadovnichiy, M. Z. Zgurovsky, Springer, 2016, 13–32 | DOI | MR

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

[14] Fomenko A. T., Nikolaenko S. S., “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

[15] Landau L. D., Lifshitz E. M., Mechanics, Butterworth–Heinemann, 1976