Integrable dynamical systems with dissipation on tangent bundles of 2D and 3D manifolds

M. V. Shamolin

Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 349-382

Résumé

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

Bibliographie

[1] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, VINITI, M., 1985

[2] Byushgens G. S., Studnev R. V., Dinamika samoleta. Prostranstvennoe dvizhenie, Mashinostroenie, M., 1988

[3] Georgievskii D. V., Shamolin M. V., “Trofimov Valerii Vladimirovich”, SMFN, 23, 2007, 5–15 | MR

[4] 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 D. V. Georgievskogo, M. V. Shamolina, S. A. Agafonova”, Geometriya i mekhanika, SMFN, 23, 2007, 16–45

[5] Georgievskii D. V., Shamolin M. V., “Zasedaniya seminara «Aktualnye problemy geometrii i mekhaniki» im. prof. V. V. Trofimova, provodyaschegosya na mekhaniko-matematicheskom fakultete MGU im. M. V. Lomonosova pod rukovodstvom S. A. Agafonova, D. V. Georgievskogo i M. V. Shamolina”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 62, 2009, 3–15 | MR

[6] Georgievskii D. V., Shamolin M. V., “Zasedaniya seminara «Aktualnye problemy geometrii i mekhaniki» im. prof. V. V. Trofimova, provodyaschegosya na mekhaniko-matematicheskom fakultete MGU im. M. V. Lomonosova pod rukovodstvom S. A. Agafonova, D. V. Georgievskogo i M. V. Shamolina”, Matematicheskaya fizika, kombinatorika i optimalnoe upravlenie, Sovrem. matem. i ee pril., 65, 2009, 3–10 | MR

[7] 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 D. V. Georgievskogo, M. V. Shamolina, S. A. Agafonova”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 76, 2012, 3–10 | MR

[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”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 88, 2013, 3–19 | Zbl

[9] 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”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 98, 2015, 3–8 | MR

[10] 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”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 100, 2016, 3–11 | MR

[11] Kozlov V. V., “Integriruemost i neintegriruemost v gamiltonovoi mekhanike”, UMN, 38:1 (1983), 3–67 | MR | Zbl

[12] Lokshin B. Ya., Samsonov V. A., Shamolin M. V., “Mayatnikovye sistemy s dinamicheskoi simmetriei”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 100, 2016, 76–133

[13] Puankare A., “Novye metody v nebesnoi mekhanike”, Puankare A. Izbrannye trudy, v. 1, 2, Nauka, M., 1971 ; 1972 | MR

[14] Trofimov V. V., Shamolin M. V., “Geometricheskie i dinamicheskie invarianty integriruemykh gamiltonovykh i dissipativnykh sistem”, Fundament. i prikl. matem., 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] Shamolin M. V., “Ob integriruemom sluchae v prostranstvennoi dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi”, Izv. RAN. MTT, 1997, no. 2, 65–68

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

[19] Shamolin M. V., “Novye integriruemye po Yakobi sluchai v dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi”, Dokl. RAN, 364:5 (1999), 627–629 | Zbl

[20] Shamolin M. V., “Polnaya integriruemost uravnenii dvizheniya prostranstvennogo mayatnika v potoke nabegayuschei sredy”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 2001, no. 5, 22–28 | Zbl

[21] Shamolin M. V., “Ob integrirovanii nekotorykh klassov nekonservativnykh sistem”, UMN, 57:1 (2002), 169–170 | DOI | MR

[22] Shamolin M. V., “Sluchai polnoi integriruemosti v prostranstvennoi dinamike tverdogo tela, vzaimodeistvuyuschego so sredoi, pri uchete vraschatelnykh proizvodnykh momenta sil po uglovoi skorosti”, Dokl. RAN, 403:4 (2005), 482–485 | MR

[23] Shamolin M. V., Metody analiza dinamicheskikh sistem s peremennoi dissipatsiei v dinamike tverdogo tela, Ekzamen, M., 2007, 352 pp.

[24] Shamolin M. V., “Polnaya integriruemost uravnenii dvizheniya prostranstvennogo mayatnika v potoke sredy pri uchete vraschatelnykh proizvodnykh momenta sily ee vozdeistviya”, Izv. RAN. MTT, 2007, no. 3, 187–192

[25] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike na kasatelnom rassloenii dvumernoi sfery”, UMN, 62:5 (2007), 169–170 | DOI | MR | Zbl

[26] Shamolin M. V., “Dinamicheskie sistemy s peremennoi dissipatsiei: podkhody, metody, prilozheniya”, Fundament. i prikl. matem., 14:3 (2008), 3–237 | MR

[27] Shamolin M. V., “Ob integriruemosti v elementarnykh funktsiyakh nekotorykh klassov dinamicheskikh sistem”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 2008, no. 3, 43–49 | Zbl

[28] Shamolin M. V., “Ob integriruemosti v elementarnykh funktsiyakh nekotorykh klassov nekonservativnykh dinamicheskikh sistem”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 62, 2009, 131–171

[29] Shamolin M. V., “Novye sluchai integriruemosti v prostranstvennoi dinamike tverdogo tela”, Dokl. RAN, 431:3 (2010), 339–343 | MR | Zbl

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

[31] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN, 444:5 (2012), 506–509 | MR

[32] 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

[33] Shamolin M. V., “Sopostavlenie sluchaev polnoi integriruemosti v dinamike dvumernogo, trekhmernogo i chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 76, 2012, 84–99 | MR | Zbl

[34] Shamolin M. V., “Mnogoobrazie sluchaev integriruemosti v prostranstvennoi dinamike tverdogo tela v nekonservativnom pole sil”, Tr. semin. im. I. G. Petrovskogo, 30, 2014, 287–350

[35] Shamolin M. V., “Novyi sluchai polnoi integriruemosti uravnenii dinamiki na kasatelnom rassloenii k trekhmernoi sfere”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 2015, no. 3, 11–14 | MR | Zbl

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

[37] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k sfere”, Probl. matem. analiza, 2016, no. 86, 139–151 | Zbl

[38] Shamolin M. V., “Transtsendentnye pervye integraly dinamicheskikh sistem na kasatelnom rassloenii s sfere”, Geometriya i mekhanika, Sovrem. matem. i ee pril., 100, 2016, 58–75

[39] 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

[40] Shamolin M. V., “Malomernye i mnogomernye mayatniki v nekonservativnom pole. Ch. 1”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 134, 2017, 6–128

[41] Shamolin M. V., “Malomernye i mnogomernye mayatniki v nekonservativnom pole. Ch. 2”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 135, 2017, 3–93

[42] Yakobi K., Lektsii po dinamike, ONTI, M.–L., 1936