Geodesic
Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 96-105.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we present tensor invariants (differential forms) for homogeneous dynamical systems on the tangent bundles of smooth three-dimensional manifolds and demonstrate the connection between the presence of these invariants and the existence of a complete set of first integrals, which is necessary for integrating geodesic, potential, and dissipative systems.
Keywords: dynamical system, integrability, dissipation, transcendental first integral, invariant differential form. 
@article{INTO_2022_210_a9,
     author = {M. V. Shamolin},
     title = {Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {96--105},
     publisher = {mathdoc},
     volume = {210},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_210_a9/}
}
TY  - JOUR
AU  - M. V. Shamolin
TI  - Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 96
EP  - 105
VL  - 210
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_210_a9/
LA  - ru
ID  - INTO_2022_210_a9
ER  - 
%0 Journal Article
%A M. V. Shamolin
%T Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 96-105
%V 210
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_210_a9/
%G ru
%F INTO_2022_210_a9
M. V. Shamolin. Some tensor invariants of geodesic, potential, and dissipative systems on the tangent bundles of three-dimensional manifolds. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 96-105. http://geodesic.mathdoc.fr/item/INTO_2022_210_a9/

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

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

[3] Veil G., Simmetriya, URSS, M., 2007

[4] 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

[5] 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

[6] 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

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

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

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

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

[11] Klein F., Neevklidova geometriya, URSS, M., 2017

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

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

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

[15] Kolmogorov A. N., “O dinamicheskikh sistemakh s integralnym invariantom na tore”, Dokl. AN SSSR., 93:5 (1953), 763–766 | MR | Zbl

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

[17] 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

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

[19] 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

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

[21] Trofimov V. V., “Simplekticheskie struktury na gruppakh avtomorfizmov simmetricheskikh prostranstv”, Vestn. Mosk. un-ta. Ser. 1. Mat. Mekh., 1984, no. 6, 31–33 | Zbl

[22] 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

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

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

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

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

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

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

[29] 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

[30] 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

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

[32] 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

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

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

[35] 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

[36] 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

[37] 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

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

[39] 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

[40] 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

[41] 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

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

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

[44] 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

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

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

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

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

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

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

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

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

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

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

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

[56] 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

[57] 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

[58] Shamolin M. V., “Novye sluchai odnorodnykh integriruemykh sistem s dissipatsiei na kasatelnom rassloenii chetyrekhmernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 497:1 (2021), 23–30 | Zbl

[59] Shamolin M. V., “Novye sluchai integriruemosti geodezicheskikh, potentsialnykh i dissipativnykh sistem na kasatelnom rassloenii konechnomernogo mnogoobraziya”, Dokl. RAN. Mat. inform. protsessy upravl., 500:1 (2021), 78–86 | Zbl

[60] Poincaré H., Calcul des probabilités, Gauthier-Villars, Paris, 1912 | MR

[61] Shamolin M. V., “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium”, J. Math. Sci., 110:2 (2002), 2528–2557 | DOI | MR | Zbl

[62] 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