Examples of nine-order integrable dynamical systems with dissipation

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 187 (2020), pp. 68-81.

Résumé

In this paper, the integrability of some classes of homogeneous with respect to a part of variables ninth-order dynamical systems, in which a system on the tangent bundle to four-dimensional manifolds is distinguished. In this case, force fields have a dissipation of different signs and are generalizations of those considered earlier.

Détail

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

Bibliographie

[1] Georgievskii D. V., Shamolin M. V., “Obobschennye dinamicheskie uravneniya Eilera dlya tverdogo tela s nepodvizhnoi tochkoi v $\mathbb{R}^n$”, Dokl. RAN., 383:5 (2002), 635–637

[2] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya, Nauka, M., 1979 | MR

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

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

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

[6] Kozlov V. V., “Ratsionalnye integraly kvaziodnorodnykh dinamicheskikh sistem”, Prikl. mat. mekh., 79:3 (2015), 307–316 | Zbl

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

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

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

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

[11] Shamolin M. V., “Klassifikatsiya fazovykh portretov v zadache o dvizhenii tela v soprotivlyayuscheisya srede pri nalichii lineinogo dempfiruyuschego momenta”, Prikl. mat. mekh., 57:4 (1993), 40–49 | MR | Zbl

[12] Shamolin M. V., “Vvedenie v zadachu o tormozhenii tela v soprotivlyayuscheisya srede i novoe dvukhparametricheskoe semeistvo fazovykh portretov”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 4 (1996), 57–69 | MR | Zbl

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

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

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

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

[17] 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

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

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

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

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

[22] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii mnogomernogo mnogoobraziya”, Dokl. RAN., 482:5 (2018), 527–533 | MR

[23] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya”, Dokl. RAN., 479:3 (2018), 270–276 | MR

[24] Shamolin M. V., “Novye sluchai integriruemykh sistem pyatogo poryadka s dissipatsiei”, Dokl. RAN., 485:5 (2019), 583–587

[25] Shamolin M. V., “Novye sluchai integriruemykh sistem sedmogo poryadka s dissipatsiei”, Dokl. RAN., 487:4 (2019), 381–386