Realizing integrable Hamiltonian systems by means of billiard books

V. A. Kibkalo ; A. T. Fomenko ; I. S. Kharcheva

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 82 (2021) no. 1, pp. 45-78.

Bibliographie

[1] S. V. Bolotin, “Integriruemye bilyardy Birkgofa”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 1990, no. 2, 33–36

[2] A. V. Bolsinov, P. Kh. Rikhter, A. T. Fomenko, “Metod krugovykh molekul i topologiya volchka Kovalevskoi”, Matem. sb., 191:2 (2000), 3–42 | Zbl

[3] A. V. Bolsinov, A. T. Fomenko, Integriruemye gamiltonovy sistemy. Geometriya. Topologiya. Klassifikatsiya, v. 1, «Udmurtskii un-t», Izhevsk, 1999

[4] V. V. Vedyushkina, “Integriruemye billiardy realizuyut toricheskie sloeniya na linzovykh prostranstvakh i 3-tore”, Matem. sb., 211:2 (2020), 46–73 | MR | Zbl

[5] V. V. Vedyushkina, “Sloenie Liuvillya bilyardnoi knizhki, modeliruyuschei sluchai Goryacheva–Chaplygina”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 1, 64–68 | MR | Zbl

[6] V. V. Vedyushkina, V. A. Kibkalo, “Realizatsiya bilyardami chislovogo invarianta rassloeniya Zeiferta integriruemykh sistem”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 4, 22–28 | MR | Zbl

[7] V. V. Vedyushkina, V. A. Kibkalo, A. T. Fomenko, “Topologicheskoe modelirovanie integriruemykh sistem billiardami: realizatsiya chislovykh invariantov”, Dokl. RAN, 493 (2020), 9–12 | Zbl

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

[9] V. V. Vedyushkina (Fokicheva), A. T. Fomenko, “Integriruemye geodezicheskie potoki na orientiruemykh dvumernykh poverkhnostyakh i topologicheskie billiardy”, Izv. RAN. Ser. matem., 83:6 (2019), 63–103 | MR | Zbl

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

[11] V. V. Vedyushkina, I. S. Kharcheva, “Billiardnye knizhki modeliruyut vse trekhmernye bifurkatsii integriruemykh gamiltonovykh sistem”, Matem. sb., 209:12 (2018), 17–56 | MR | Zbl

[12] A. A. Glutsyuk, “O dvumernykh polinomialno integriruemykh bilyardakh na poverkhnostyakh postoyannoi krivizny”, Dokl. RAN, 481:6 (2018), 594–598 | Zbl

[13] V. I. Dragovich, M. Radnovich, “Psevdointegriruemye billiardy i reshetki dvoinykh otrazhenii”, UMN, 70:1 (421) (2015), 3–34 | MR

[14] E. O. Kantonistova, “Topologicheskaya klassifikatsiya integriruemykh gamiltonovykh sistem na poverkhnostyakh vrascheniya v potentsialnom pole”, Matem. sb., 207:3 (2016), 47–92 | MR | Zbl

[15] V. A. Kibkalo, “Svoistvo nekompaktnosti sloev i osobennostei neevklidovoi sistemy Kovalevskoi na puchke algebr Li”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2000, no. 6, 56–59

[16] V. A. Kibkalo, “Topologicheskaya klassifikatsiya sloenii Liuvillya dlya integriruemogo sluchaya Kovalevskoi na algebre Li $\mathrm{so}(4)$”, Matem. sb., 210:5 (2019), 3–40 | MR | Zbl

[17] V. A. Kibkalo, “Billiardy s potentsialom modeliruyut ryad chetyrekhmernykh osobennostei integriruemykh sistem”, Sovr. prob. matem. i mekh., Mater. mezhd. konf., posv. 80-letiyu akad. RAN V. A. Sadovnichego (Moskva, 2019), v. 2, 563–566

[18] V. A. Kibkalo, “Topologiya analoga sluchaya integriruemosti Kovalevskoi na algebre Li $\mathrm{so}(4)$ pri nulevoi postoyannoi ploschadei”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 4, 46–50 | MR

[19] I. F. Kobtsev, “Ellipticheskii billiard v pole potentsialnykh sil: klassifikatsiya dvizhenii, topologicheskii analiz”, Matem. sb., 211:7 (2020), 93–120 | MR | Zbl

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

[21] I. K. Kozlov, A. A. Oshemkov, “Klassifikatsiya osobennostei tipa sedlo-fokus”, Chebyshevskii sb., 21:2 (2020), 228–243 | MR | Zbl

[22] E. A. Kudryavtseva, “Integriruemye po Liuvillyu obobschennye billiardnye potoki i teoremy tipa Ponsele”, Fundam. Prikl. Matem., 20:3 (2015), 113–152

[23] E. A. Kudryavtseva, A. A. Oshemkov, “Bifurkatsii integriruemykh mekhanicheskikh sistem s magnitnym polem na poverkhnostyakh vrascheniya”, Chebyshevskii sb., 21:2 (2020), 244–265 | MR | Zbl

[24] P. V. Morozov, “Topologiya sloenii Liuvillya sluchaev integriruemosti Steklova i Sokolova uravnenii Kirkhgofa”, Matem. sb., 195:3 (2004), 69–114 | MR

[25] V. A. Moskvin, “Topologiya sloenii Liuvillya integriruemogo bilyarda v nevypuklykh oblastyakh”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2018, no. 3, 21–29 | Zbl

[26] A. A. Oshemkov, “Funktsii Morsa na dvumernykh poverkhnostyakh. Kodirovanie osobennostei”, Novye rezultaty v teorii topologicheskoi klassifikatsii integriruemykh sistem, Sbornik statei, Tr. MIAN, 205, Nauka, M., 1994, 131–140

[27] S. E. Pustovoitov, “Topologicheskii analiz billiarda v ellipticheskom koltse v potentsialnom pole”, Fundament. i prikl. matem., 22:6 (2019), 201–225 | MR

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

[29] V. V. Fokicheva, A. T. Fomenko, “Integriruemye billiardy modeliruyut vazhnye integriruemye sluchai dinamiki tverdogo tela”, Dokl. RAN, 465:2 (2015), 1–4

[30] A. T. Fomenko, “Teoriya Morsa integriruemykh gamiltonovykh sistem”, DAN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[31] A. T. Fomenko, “Topologiya poverkhnostei postoyannoi energii nekotorykh integriruemykh gamiltonovykh sistem i prepyatstviya k integriruemosti”, Izv. AN SSSR. Ser. matem., 50:6 (1986), 1276–1307 | MR | Zbl

[32] A. T. Fomenko, V. V. Vedyushkina, “Bilyardy i integriruemost v geometrii i fizike. Novyi vzglyad i novye vozmozhnosti”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2019, no. 3, 15–25 | Zbl

[33] A. T. Fomenko, Kh. Tsishang, “O topologii trekhmernykh mnogoobrazii, voznikayuschikh v gamiltonovoi mekhanike”, DAN SSSR, 294:2 (1987), 283–287 | MR | Zbl

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

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

[36] I. S. Kharcheva, “Izoenergeticheskie mnogoobraziya integriruemykh bilyardnykh knizhek”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2020, no. 4, 12–22 | MR | Zbl

[37] A. Avila, J. De Simoi, V. Kaloshin, “An integrable deformation of an ellipse of small eccentricity is an ellipse”, Ann. of Math. (2), 184:2 (2016), 527–558 | DOI | MR | Zbl

[38] M. Bialy, A. E. Mironov, “Angular billiard and algebraic Birkhoff conjecture”, Adv. Math., 313 (2017), 102–126 | DOI | MR | Zbl

[39] M. Bialy, A. E. Mironov, “Algebraic Birkhoff conjecture for billiards on sphere and hyperbolic plane”, J. Geom. Phys., 115 (2017), 150–156 | DOI | MR | Zbl

[40] A. V. Bolsinov, A. A. Oshemkov, “Singularities of integrable Hamiltonian systems”, Topological methods in the theory of integrable systems, Camb. Sci. Publ., Cambridge, 2006, 1–67 | MR | Zbl

[41] A. T. Fomenko, V. A. Kibkalo, “Saddle singularities in integrable Hamiltonian systems: Examples and algorithms”, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, Understanding Complex Systems, eds. V. A. Sadovnichiy, M. Z. Zgurovsky, Springer, 2021, 3–26 | DOI | MR | Zbl

[42] A. T. Fomenko, V. V. Vedyushkina, “Implementation of Integrable Systems by Topological, Geodesic Billiards with Potential and Magnetic Field”, Russ. J. Math. Phys., 26 (2019), 320–333 | DOI | MR | Zbl

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

[44] V. A. Kibkalo, “Topological analysis of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra $\mathrm{so}(4)$”, Lobachevskii J. Math., 39:9 (2018), 1396–1399 | DOI | MR | Zbl

[45] V. A. Kibkalo, “Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra $\mathrm{so}(3,1)$”, Topol. Appl., 275 (2020), 107028 | DOI | MR | Zbl

[46] V. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Springer, Berlin, 1993 | MR | Zbl

[47] A. A. Oshemkov, “Fomenko invariants for the main integrable cases of the rigid body motion equations”, Topological classification of integrable systems, Adv. Soviet Math., 6, AMS, Providence, RI, 1991, 67–146 | MR

[48] V. V. Vedyushkina, I. S. Kharcheva, “Billiard books realize all bases of Liouville foliations of integrable Hamiltonian systems”, Sb. Math., 212 (2021) | DOI | MR

[49] F. Waldhausen, “Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. I”, Invent. Math., 3:4 (1967), 308–333 | DOI | MR | Zbl

[50] F. Waldhausen, “Eine Klasse von 3-dimensionalen Mannigfaltighkeiten. II”, Invent. Math., 4:2 (1967), 88–117 | DOI | MR

[51] N. T. Zung, “Symplectic topology of integrable Hamiltonian systems. I: Arnold–Liouville with singularities”, Compositio Math., 101:2 (1996), 179–215 | MR | Zbl