Geodesic
Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2014), pp. 18-27 
V. V. Fokicheva. Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2014), pp. 18-27. http://geodesic.mathdoc.fr/item/VMUMM_2014_4_a2/
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     author = {V. V. Fokicheva},
     title = {Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2014_4_a2/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

An integrable system that is a billiard in a domain bounded by confocal ellipses and hyperbolas is studied. This system arises in the description of the motion of a point inside this domain with natural reflection from the boundary. The topological invariant of Liouville equivalence of such systems, namely, the Fomenko–Zieschang molecule, is calculated using a new method developed by the author.

[1] Kozlov V.V., Treschev D.V., Billiardy. Geneticheskoe vvedenie v dinamiku sistem s udarami, Izd-vo MGU, M., 1991 | MR

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

[3] Birkgof Dzh.D., Dinamicheskie sistemy, Izdatelskii dom “Udmurtskii universitet”, Izhevsk, 1999

[4] Bolsinov A.V., Fomenko A.T., “Traektornaya klassifikatsiya geodezicheskikh potokov dvumernykh ellipsoidov. Zadacha Yakobi traektorno ekvivalentna integriruemomu sluchayu Eilera v dinamike tverdogo tela”, Funkts. analiz i ego pril., 29:3 (1995), 1–15 | MR | Zbl

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

[6] Dragovich V., Radnovich M., Integriruemye billiardy, kvadriki i mnogomernye porizmy Ponsele, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.–Izhevsk, 2010

[7] Arnold V.I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1989 | MR

[8] Fomenko A.T., Tsishang Kh., “O tipichnykh topologicheskikh svoistvakh integriruemykh gamiltonovykh sistem”, Izv. AN SSSR. Ser. matem., 52:2 (1988), 378–407 | Zbl

[9] Fomenko A.T., “Simplekticheskaya topologiya vpolne integriruemykh gamiltonovykh sistem”, Uspekhi matem. nauk, 44:1(265) (1989), 145–173 | MR

[10] Fomenko A.T., Tsishang Kh., “Topologicheskii invariant i kriterii ekvivalentnosti integriruemykh gamiltonovykh sistem s dvumya stepenyami svobody”, Izv. AN SSSR. Ser. matem., 54:3 (1990), 546–575 | Zbl

[11] Gutkin E., “Billiard dynamics: a survey with the emphasis on open problems”, Regul. Chaotic Dyn., 8:1 (2003), 1–13 | DOI | MR | Zbl

[12] Fokicheva V.V., “Opisanie osobennostei sistemy “bilyard v ellipse””, Vestn. Mosk. un-ta. Matem. Mekhan., 2012, no. 5, 31–34 | MR | Zbl

[13] Kudryavtseva E.A., Nikonov I.M., Fomenko A.T., “Maksimalno simmetrichnye kletochnye razbieniya poverkhnostei i ikh nakrytiya”, Matem. sb., 199:9 (2008), 3–96 | DOI | MR | Zbl