Integrable homogeneous dynamical systems with dissipation on the tangent bundle of a three-dimensional manifold

M. V. Shamolin

Geodesic

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Integrable homogeneous dynamical systems with dissipation on the tangent bundle of a three-dimensional manifold

M. V. Shamolin

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 205 (2022), pp. 22-54.

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Résumé

In many problems of dynamics, the position spaces of systems considered are three-dimensional manifolds and hence the phase spaces of such systems are the corresponding tangent bundles. In this paper possess, we consider dynamical systems with variable (alternating) dissipation. We prove the integrability of general classes of homogeneous dynamical systems with variable dissipation on the tangent bundles of three-dimensional manifolds.

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

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M. V. Shamolin. Integrable homogeneous dynamical systems with dissipation on the tangent bundle of a three-dimensional manifold. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 205 (2022), pp. 22-54. http://geodesic.mathdoc.fr/item/INTO_2022_205_a3/

Bibliographie
Cité par

[1] Bendikson I., “O krivykh, opredelyaemykh differentsialnymi uravneniyami”, Usp. mat. nauk., 9 (1941), 119–211

[2] Bogoyavlenskii O. I., “Dinamika tverdogo tela s $n$ ellipsoidalnymi polostyami, zapolnennymi magnitnoi zhidkostyu”, Dokl. AN SSSR., 272:6 (1983), 1364–1367 | MR | Zbl

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

[4] Burbaki N., Integrirovanie. Mery, integrirovanie mer, Nauka, M., 1967 | MR

[5] Burbaki N., Integrirovanie. Vektornoe integrirovanie. Mera Khaara. Svertka i predstavleniya, Nauka, M., 1970

[6] Burbaki N., Integrirovanie. Mery na lokalno kompaktnykh prostranstvakh. Prodolzhenie mery. Integrirovanie mer. Mery na otdelimykh prostranstvakh, Nauka, M., 1977

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

[8] Georgievskii D. V., Shamolin M. V., “Kinematika i geometriya mass tverdogo tela s nepodvizhnoi tochkoi v $\mathbb{R}^n$”, Dokl. RAN., 380:1 (2001), 47–50

[9] 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

[10] Georgievskii D. V., Shamolin M. V., “Pervye integraly uravnenii dvizheniya obobschennogo giroskopa v $\mathbb{R}^{n}$”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 5 (2003), 37–41 | Zbl

[11] Georgievskii D. V., Shamolin M. V., “Simvoly Levi-Chivity, obobschennye vektornye proizvedeniya i novye sluchai integriruemosti v mekhanike mnogomernogo tela”, Sovr. mat. prilozh., 76 (2012), 22–39

[12] Golubev V. V., Lektsii po integrirovaniyu uravnenii dvizheniya tyazhelogo tverdogo tela okolo nepodvizhnoi tochki, Gostekhizdat, M.-L., 1953 | MR

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

[14] Eroshin V. A., Samsonov V. A., Shamolin M. V., “Modelnaya zadacha o tormozhenii tela v soprotivlyayuscheisya srede pri struinom obtekanii”, Izvyu RAN. Mekh. zhidk. gaza., 1995, no. 3, 23–27

[15] Ivanova T. A., “Ob uravneniyakh Eilera v modelyakh teoreticheskoi fiziki”, Mat. zametki., 52:2 (1992), 43–51

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

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

[18] Kozlov V. V., “Tenzornye invarianty i integrirovanie differentsialnykh uravnenii”, Usp. mat. nauk., 74:1 (445) (2019), 117–148 | MR | Zbl

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

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

[21] Pokhodnya N. V., Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tela”, Vestn. SamGU. Estestvennonauch. ser., 9:100 (2012), 136–150 | Zbl

[22] Pokhodnya N. V., Shamolin M. V., “Nekotorye usloviya integriruemosti dinamicheskikh sistem v transtsendentnykh funktsiyakh”, Vestn. SamGU. Estestvennonauch. ser., 9/1:110 (2013), 35–41 | Zbl

[23] Pokhodnya N. V., Shamolin M. V., “Integriruemye sistemy na kasatelnom rassloenii k mnogomernoi sfere”, Vestn. SamGU. Estestvennonauch. ser., 7:118 (2014), 60–69 | Zbl

[24] Samsonov V. A., Shamolin M. V., “K zadache o dvizhenii tela v soprotivlyayuscheisya srede”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 1989, no. 3, 51–54 | Zbl

[25] Tikhonov A. A., “Metod upravleniya dlya uglovoi stabilizatsii elektrodinamicheskoi trosovoi sistemy”, Avtomat. telemekh., 2020, no. 2, 91–114 | Zbl

[26] Trofimov V. V., “Uravneniya Eilera na konechnomernykh razreshimykh gruppakh Li”, Izv. AN SSSR. Ser. mat., 44:5 (1980), 1191–1199 | MR | Zbl

[27] Trofimov V. V., Fomenko A. T., “Metodika postroeniya gamiltonovykh potokov na simmetricheskikh prostranstvakh i integriruemost nekotorykh gidrodinamicheskikh sistem”, Dokl. AN SSSR., 254:6 (1980), 1349–1353 | MR

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

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

[30] Shabat B. V., Vvedenie v kompleksnyi analiz, Nauka, M., 1987 | MR

[31] Shamolin M. V., “K zadache o dvizhenii tela v srede s soprotivleniem”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 1 (1992), 52–58 | MR | Zbl

[32] 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

[33] 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

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

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

[36] Shamolin M. V., “Integriruemost po Yakobi v zadache o dvizhenii chetyrekhmernogo tverdogo tela v soprotivlyayuscheisya srede”, Dokl. RAN., 375:3 (2000), 343–346

[37] Shamolin M. V., “Ob integrirovanii nekotorykh klassov nekonservativnykh sistem”, Usp. mat. nauk., 57:1 (2002), 169–170 | MR

[38] Shamolin M. V., “Ob odnom integriruemom sluchae uravnenii dinamiki na $\textrm{so}(4)\times\mathbf{R}^{4}$”, Usp. mat. nauk., 60:6 (2005), 233–234 | MR | Zbl

[39] Shamolin M. V., “Sopostavlenie integriruemykh po Yakobi sluchaev ploskogo i prostranstvennogo dvizheniya tela v srede pri struinom obtekanii”, Prikl. mat. mekh., 69:6 (2005), 1003–1010 | MR | Zbl

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

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

[42] Shamolin M. V., “Novye sluchai polnoi integriruemosti v dinamike dinamicheski simmetrichnogo chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 425:3 (2009), 338–342 | MR | Zbl

[43] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrekhmernogo tverdogo tela v nekonservativnom pole”, Usp. mat. nauk., 65:1 (2010), 189–190 | MR | Zbl

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

[45] Shamolin M. V., “Polnyi spisok pervykh integralov v zadache o dvizhenii chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 440:2 (2011), 187–190 | MR

[46] 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

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

[48] Shamolin M. V., “Novyi sluchai integriruemosti uravnenii dinamiki na kasatelnom rassloenii k trekhmernoi sfere”, Usp. mat. nauk., 68:5 (413) (2013), 185–186 | MR | Zbl

[49] Shamolin M. V., “Polnyi spisok pervykh integralov dinamicheskikh uravnenii dvizheniya chetyrekhmernogo tverdogo tela v nekonservativnom pole pri nalichii lineinogo dempfirovaniya”, Dokl. RAN., 449:4 (2013), 416–419 | MR

[50] Shamolin M. V., “Novyi sluchai integriruemosti v dinamike mnogomernogo tverdogo tela v nekonservativnom pole pri uchete lineinogo dempfirovaniya”, Dokl. RAN., 457:5 (2014), 542–545 | MR

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

[52] Shamolin M. V., “Polnyi spisok pervykh integralov dinamicheskikh uravnenii dvizheniya mnogomernogo tverdogo tela v nekonservativnom pole”, Dokl. RAN., 461:5 (2015), 533–536 | MR

[53] 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 | MR

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

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

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

[57] Shamolin M. V., “Novye sluchai integriruemykh sistem s dissipatsiei na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Dokl. RAN., 477:2 (2017), 168–172 | MR

[58] Shamolin M. V., “Integriruemye dinamicheskie sistemy s konechnym chislom stepenei svobody s dissipatsiei”, Probl. mat. anal., 2018, no. 95, 79–101 | MR | Zbl

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

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

[61] Shamolin M. V., “Novye sluchai integriruemykh sistem devyatogo poryadka s dissipatsiei”, Dokl. RAN., 489:6 (2019), 592–598

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

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

[64] Shamolin M. V., “Novye sluchai integriruemykh sistem nechetnogo poryadka s dissipatsiei”, Dokl. RAN. Mat. inform. protsessy upravl., 491:1 (2020), 95–101 | MR | Zbl

[65] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii dvumernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 494:1 (2020), 105–111 | MR | Zbl

[66] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii trekhmernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 495:1 (2020), 84–90 | MR | Zbl

[67] Shamolin M. V., Integriruemye dinamicheskie sistemy s dissipatsiei. Kn. 1. Tverdoe telo v nekonservativnom pole, LENAND, M., 2019

[68] Shamolin M. V., Integriruemye dinamicheskie sistemy s dissipatsiei. Kn. 2: Zakreplennye mayatniki raznoi razmernosti, LENAND, M., 2021

[69] Aleksandrov A. Y., Aleksandrova E. B., Tikhonov A. A., “On the monoaxial stabilization of a rigid body under vanishing restoring torque”, AIP Conf. Proc., 1959 (2018), 080001 | DOI | MR

[70] Tikhonov A. A., Yakovlev A. B., “On dependence of equilibrium characteristics of the space tethered system on environmental parameters”, Int. J. Plasma Env. Sci. Techn., 13:1, 49–52 | MR