Examples of Integrable Systems with Dissipation on the Tangent Bundles of Multidimensional Spheres

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 150 (2018), pp. 78-87.

Résumé

In this paper, we prove the integrability of certain classes of dynamical systems that appear in the dynamics of multidimensional rigid bodies and the dynamics of a particle moving on a multidimensional sphere. Force field considered have the so-called variable dissipation with zero mean; they are generalizations of fields studied earlier. We present examples of the application of the method for integrating dissipative systems on the tangent bundles of two-dimensional surfaces of revolution.

Détail

Keywords: dynamical system, nonconservative force field, integrability, transcendental first integral.

Bibliographie

[6] Bogoyavlenskii O. I., “Nekotorye integriruemye sluchai uravnenii Eilera”, Dokl. AN SSSR, 287:5 (1986), 1105–1108 | MR

[7] Veselov A. P., “Ob usloviyakh integriruemosti uravnenii Eilera na $\textrm{so}(4)$”, Dokl. AN SSSR, 270:6 (1983), 1298–1300 | MR | Zbl

[8] Georgievskii D. V., Shamolin M. V., “Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova «Aktualnye problemy geometrii i mekhaniki» im. prof. V. V. Trofimova pod rukovodstvom S. A. Agafonova, D. V. Georgievskogo i M. V. Shamolina”, Sovr. mat. prilozh., 100 (2016), 3–11 | MR

[9] Dubrovin B. A., Novikov S. P., “O skobkakh Puassona gidrodinamicheskogo tipa”, Dokl. AN SSSR, 219:2 (1984), 228–237

[10] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya. Metody i prilozheniya, Nauka, M., 1979 | MR

[11] Kamke E., Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1971

[12] Lokshin B. Ya., Samsonov V. A., Shamolin M. V., “Mayatnikovye sistemy s dinamicheskoi simmetriei”, Sovr. mat. prilozh., 100 (2016), 76–133

[13] Manakov S. V., “Zamechanie ob integrirovanii uravnenii Eilera dinamiki $n$-mernogo tverdogo tela”, Funkts. anal. prilozh., 10:4 (1976), 93–94 | MR | Zbl

[14] Trofimov V. V., Shamolin M. V., “Geometricheskie i dinamicheskie invarianty integriruemykh gamiltonovykh i dissipativnykh sistem”, Fundam. prikl. mat., 16:4 (2010), 3–229

[15] Chaplygin S. A., “O dvizhenii tyazhelykh tel v neszhimaemoi zhidkosti”, Poln. sobr. soch., v. 1, Izd-vo AN SSSR, L., 1933, 133–135

[16] Chaplygin S. A., Izbrannye trudy, Nauka, M., 1976 | MR

[17] Shabat B. V., Vvedenie v kompleksnyi analiz, Nauka, M., 1987

[18] Shamolin M. V., “Ob integriruemosti v transtsendentnykh funktsiyakh”, Usp. mat. nauk, 53:3 (1998), 209–210 | DOI | MR | Zbl

[19] Shamolin M. V., “Dinamicheskie sistemy s peremennoi dissipatsiei: podkhody, metody, prilozheniya”, Fundam. prikl. mat., 14:3 (2008), 3–237

[20] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN, 437:2 (2011), 190–193 | MR

[21] Shamolin M. V., “Novyi sluchai integriruemosti v prostranstvennoi dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi, pri uchete lineinogo dempfirovaniya”, Dokl. RAN, 442:4 (2012), 479–481 | MR

[22] Shamolin M. V., “Mnogoobrazie sluchaev integriruemosti v dinamike malomernogo i mnogomernogo tverdogo tela v nekonservativnom pole sil”, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obzory, 125 (2013), 5–254

[23] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN, 453:1 (2013), 46–49 | DOI | MR

[24] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere i prilozheniya”, Fundam. prikl. mat., 20:4 (2015), 3–231 | MR

[25] Shamolin M. V., “Polnyi spisok pervykh integralov uravnenii dvizheniya mnogomernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN, 464:6 (2015), 688–692 | DOI | MR

[26] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere i prilozheniya”, Fundam. prikl. mat., 20:4 (2015), 3–231 | MR

[27] Shamolin M. V., “Novye sluchai integriruemosti sistem s dissipatsiei na kasatelnykh rassloeniyakh k dvumernoi i trekhmernoi sferam”, Dokl. RAN, 471:5 (2016), 547–551 | DOI | MR

[28] Shamolin M. V., “Chetyrekhmernoe tverdoe telo-mayatnik v nekonservativnom pole”, Mat. mezhdunar. konf. «Voronezhskaya zimnyaya matematicheskaya shkola S. G. Kreina–2016», Voronezh, 2016, 433–436

[29] Shamolin M. V., “Integriruemye sistemy v dinamike na kasatelnom rassloenii k sfere”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 2016, no. 2, 25–30 | Zbl

[30] Shamolin M. V., “Integriruemye nekonservativnye dinamicheskie sistemy na kasatelnom rassloenii k mnogomernoi sfere”, Differ. uravn., 52:6 (2016), 743–759 | DOI | Zbl

[31] Shamolin M. V., “Pervye integraly dinamicheskikh sistem s peremennoi dissipatsiei v dinamike tverdogo tela”, Ustoichivost i kolebaniya nelineinykh sistem upravleniya, Mat. XIII Mezhdunar. konf., Moskva, 1–3 iyunya 2016 g., M., 2016, 421–423

[32] Shamolin M. V., “Pervye integraly dinamicheskikh sistem s dissipatsiei na kasatelnom rassloenii konechnomernoi sfery”, Geometricheskii analiz i ego prilozheniya, Mat. III Mezhdunar. shkoly-konf., Volgograd, 30 maya–3 iyunya 2016 g., Volgograd, 2016, 217–222

[33] Shamolin M. V., “Mnogomernyi mayatnik v nekonservativnom silovom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN, 470:3 (2016), 288–292 | DOI | MR

[34] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k sfere”, Probl. mat. anal., 86 (2016), 139–151 | Zbl

[35] Shamolin M. V., “Sluchai integriruemosti, sootvetstvuyuschie dvizheniyu mayatnika na dvumernoi ploskosti”, Sovr. mat. prilozh., 100 (2016), 36–57

[36] Shamolin M. V., “Transtsendentnye pervye integraly dinamicheskikh sistem na kasatelnom rassloenii k sfere”, Sovr. mat. prilozh., 100 (2016), 58–75

[37] Shamolin M. V., “Sluchai integriruemosti, sootvetstvuyuschie dvizheniyu mayatnika v trekhmernom prostranstve”, Vestn. SamGU. Estestvennonauch. ser., 2016, no. 3-4, 75–97 | Zbl

[38] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere”, Dokl. RAN, 474:2 (2017), 177–181 | DOI | MR

[39] Shamolin M. V., “Malomernye i mnogomernye mayatniki v nekonservativnom pole. Ch. 1”, Itogi nauki i tekhn. Ser. Sovr. mat. i ee prilozh. Tematich. obz., 134 (2017), 6–128

[40] Shamolin M. V., “Malomernye i mnogomernye mayatniki v nekonservativnom pole. Ch. 2”, Itogi nauki i tekhn. Ser. Sovr. mat. i ee prilozh. Tematich. obz., 135 (2017), 3–93

[41] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii k mnogomernoi sfere”, Itogi nauki i tekhn. Ser. Sovr. mat. i ee prilozh. Tematich. obz., 137 (2017), 104–117

[42] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii dvumernogo mnogoobraziya”, «Matematicheskaya teoriya optimalnogo upravleniya», Mat. mezhdunar. konf., posvyasch. 90-letiyu akad. R. V. Gamkrelidze (Moskva, 1–2 iyunya 2017 g.), Mat. in-t im. V. A. Steklova RAN, M., 2017, 124–127

[43] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii dvumernogo mnogoobraziya”, Dokl. RAN, 475:5 (2017), 519–523 | MR

[44] Shamolin M. V., “Integrability in elementary functions of certain classes of nonconservative systems”, Proc. 7th Eur. Conf. on Applied Mathematics and Informatics (AMATHI'16) (Venice, Italy, January 29–31, 2016), WSEAS Press, Seoul, 2016, 50–58

[45] Shamolin M. V., “Cases of integrability corresponding to the motion of a pendulum in the three-dimensional space”, Proc. XLIV Summer School-Conference «Advanced Problems in Mechanics» Dedicated to the 30th Anniversary of IPME RAS (St. Petersburg, June 27–July 2, 2016), St. Petersburg, 2016, 375–387

[46] Shamolin M. V., “Cases of integrability corresponding to the motion of a pendulum in the three-dimensional space”, Proc. Global Conference on Applied Physics and Mathematics (Rome, July 25–27, 2016), Rome, 2016 | MR

[47] Shamolin M. V., “First integrals of variable dissipation dynamical systems in rigid body dynamics”, Int. Conf. «Stability and Oscillations of Nonlinear Control Systems» (Pyatnitskiy's Conference) (Moscow, 1–3 June 2016), IEEE, 2016, 1–4

[48] Shamolin M. V., “Cases of integrability corresponding to the motion of a pendulum in the four-dimensional space”, Proc. XLV Summer School-Conf. «Advanced Problems in Mechanics» (St. Petersburg, June 22–27, 2017), St. Petersburg, 2017, 401–413