Integrable variable dissipation systems on the tangent bundle of a~multi-dimensional sphere and some applications

M. V. Shamolin

Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 4, pp. 3-231.

Résumé

This paper is a survey of integrable cases in dynamics of a multi-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either the energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods for the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.

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[115] Shamolin M. V., “Zamknutye traektorii razlichnogo topologicheskogo tipa v zadache o dvizhenii tela v srede s soprotivleniem”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1992, no. 2, 52–56 | MR | Zbl

[116] Shamolin M. V., “K zadache o dvizhenii tela v srede s soprotivleniem”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1992, no. 1, 52–58 | MR | Zbl

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

[118] Shamolin M. V., “Primenenie metodov topograficheskikh sistem Puankare i sistem sravneniya v nekotorykh konkretnykh sistemakh differentsialnykh uravnenii”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1993, no. 2, 66–70 | MR | Zbl

[119] Shamolin M. V., “Suschestvovanie i edinstvennost traektorii, imeyuschikh v kachestve predelnykh mnozhestv beskonechno udalënnye tochki, dlya dinamicheskikh sistem na ploskosti”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1993, no. 1, 68–71 | MR | Zbl

[120] Shamolin M. V., “Novoe dvuparametricheskoe semeistvo fazovykh portretov v zadache o dvizhenii tela v srede”, Dokl. RAN, 337:5 (1994), 611–614 | MR | Zbl

[121] Shamolin M. V., “Ob otnositelnoi grubosti dinamicheskikh sistem v zadache o dvizhenii tela v soprotivlyayuscheisya srede: Tez. dokl. Chebyshëvskikh chtenii”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1995, no. 6, 17

[122] Shamolin M. V., “Otnositelnaya strukturnaya ustoichivost dinamicheskikh sistem zadachi dvizheniya tela v srede”, Analiticheskie, chislennye i eksperimentalnye metody v mekhanike, Sb. nauch. trudov, eds. B. E. Pobedrya, V. V. Kozlov, Izd-vo Mosk. un-ta, M., 1995, 14–19 | MR

[123] Shamolin M. V., “Vvedenie v zadachu o tormozhenii tela v soprotivlyayuscheisya srede i novoe dvukhparametricheskoe semeistvo fazovykh portretov”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 1996, no. 4, 57–69 | MR | Zbl

[124] Shamolin M. V., “Vvedenie v prostranstvennuyu dinamiku dvizheniya tvërdogo tela v soprotivlyayuscheisya srede”, Mater. mezhdunar. konf. i Chebyshëvskikh chtenii, posvyasch. 175-letiyu so dnya rozhd. P. L. Chebysheva (Moskva, 14–19 maya 1996 g.), v. 2, M., 1996, 371–373

[125] Shamolin M. V., “Mnogoobrazie tipov fazovykh portretov v dinamike tvërdogo tela, vzaimodeistvuyuschego s soprotivlyayuscheisya sredoi”, Dokl. RAN, 349:2 (1996), 193–197 | MR | Zbl

[126] Shamolin M. V., “Ob odnom integriruemom sluchae v dinamike prostranstvennogo dvizheniya tela v soprotivlyayuscheisya srede”, II Simpozium po klassicheskoi i nebesnoi mekhanike, Tez. dokl. (Velikie Luki, 23–28.08.1996), M.–Velikie Luki, 1996, 91–92

[127] Shamolin M. V., “Opredelenie otnositelnoi grubosti i dvuparametricheskoe semeistvo fazovykh portretov v dinamike tvërdogo tela”, UMN, 51:1 (1996), 175–176 | DOI | MR | Zbl

[128] Shamolin M. V., “Periodicheskie i ustoichivye po Puassonu traektorii v zadache o dvizhenii tela v soprotivlyayuscheisya srede”, Izv. RAN. MTT, 1996, no. 2, 55–63

[129] Shamolin M. V., “Spisok integralov dinamicheskikh uravnenii v prostranstvennoi zadache o dvizhenii tela v soprotivlyayuscheisya srede”, Modelirovanie i issledovanie ustoichivosti sistem, Nauch. konf. (20–24 maya 1996 g.), Tez. dokl. (Issledovanie sistem), Kiev, 1996, 142

[130] Shamolin M. V., “Integriruemost po Yakobi zadachi o prostranstvennom mayatnike, pomeschënnom v potok nabegayuschei sredy”, Modelirovanie i issledovanie ustoichivosti sistem (Modelling and Investigation of System Stability), Nauch. konf. (19–23 maya 1997 g.), Tez. dokl. (Mechanical Systems), Kiev, 1997, 143

[131] Shamolin M. V., “Ob integriruemom sluchae v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Izv. RAN. MTT, 1997, no. 2, 65–68

[132] Shamolin M. V., “Prostranstvennaya dinamika tvërdogo tela, vzaimodeistvuyuschego so sredoi: Sem. po mekh. sistem i probl. upravleniya dvizh. i navig.”, Izv. RAN. MTT, 1997, no. 4, 174

[133] Shamolin M. V., “Prostranstvennye topograficheskie sistemy Puankare i sistemy sravneniya”, UMN, 52:3 (1997), 177–178 | DOI | MR | Zbl

[134] Shamolin M. V., “Absolyutnaya i otnositelnaya strukturnaya ustoichivost v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tr. Mezhdunar. konf. “Matematika v industrii”, ICIM-98 (Taganrog, 29 iyunya – 03 iyulya 1998 g.), Izd-vo TGPI, Taganrog, 1998, 332–333

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

[136] Shamolin M. V., “Semeistvo portretov s predelnymi tsiklami v ploskoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Izv. RAN. MTT, 1998, no. 2, 29–37

[137] Shamolin M. V., “Nekotorye klassy chastnykh reshenii v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Izv. RAN. MTT, 1999, no. 2, 178–189 | MR

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

[139] Shamolin M. V., “O grubosti dissipativnykh sistem i otnositelnoi grubosti i negrubosti sistem s peremennoi dissipatsiei”, UMN, 54:5 (1999), 181–182 | DOI | MR | Zbl

[140] Shamolin M. V., “Zadacha o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede i odin sluchai integriruemosti”, Book of Abs. Third Int. Conf. “Differential Equations and Applications” (Saint-Petersburg, Russia, June 12–17, 2000), Izd-vo SPbGTU, 2000, 198

[141] Shamolin M. V., “Integriruemost po Yakobi v zadache o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede”, Dokl. RAN, 375:3 (2000), 343–346 | MR

[142] Shamolin M. V., “Integriruemost po Yakobi zadachi o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 21–26 avgusta 2000 g.), Vladimir. gos. un-t, Vladimir, 2000, 196–197

[143] Shamolin M. V., “Novoe semeistvo fazovykh portretov v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Dokl. RAN, 371:4 (2000), 480–483 | MR

[144] Shamolin M. V., “Ob odnom sluchae integriruemosti po Yakobi v dinamike chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. Mezhdunar. konf. po differents. i integr. uravn. (Odessa, 12–14 sentyabrya 2000 g.), AstroPrint, Odessa, 2000, 294–295

[145] Shamolin M. V., “O grubosti dissipativnykh sistem i otnositelnoi grubosti sistem s peremennoi dissipatsiei: Tez. dokl. sem. po vekt. i tenz. anal. im. P. K. Rashevskogo”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2000, no. 2, 63

[146] Shamolin M. V., “O predelnykh mnozhestvakh differentsialnykh uravnenii okolo singulyarnykh osobykh tochek”, UMN, 55:3 (2000), 187–188 | DOI | MR | Zbl

[147] Shamolin M. V., “Integriruemost zadachi o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede: Zasedaniya seminara ‘Aktualnye problemy geometrii i mekhaniki’ ”, Fundament. i prikl. matem., 7:1 (2001), 309

[148] Shamolin M. V., “Novye integriruemye po Yakobi sluchai v dinamike dvumernogo, trëkhmernogo i chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Ann. dokl. VIII Vseross. s'ezda po teoret. i prikl. mekhan. (Perm, 23–29 avgusta 2001 g.), UrO RAN, Ekaterinburg, 2001, 599–600

[149] Shamolin M. V., “Novye integriruemye sluchai v dinamike chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Nauch. konf. (22–25 maya 2001 g.), Thes. of Conf. Rep., Kiev, 2001, 344

[150] Shamolin M. V., “Ob ustoichivosti dvizheniya tvërdogo tela v soprotivlyayuscheisya srede, zakruchennogo vokrug svoei prodolnoi osi”, Izv. RAN. MTT, 2001, no. 1, 189–193

[151] 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 | MR | Zbl

[152] Shamolin M. V., “Sluchai integriruemosti uravnenii prostranstvennoi dinamiki tvërdogo tela”, Prikl. mekh., 37:6 (2001), 74–82 | MR | Zbl

[153] Shamolin M. V., “Novye integriruemye sluchai v dinamike dvukhmernogo, trëkhmernogo i chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 1–6 iyulya 2002 g.), Vladimir. gos. un-t, Vladimir, 2002, 142–144

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

[155] Shamolin M. V., “Geometricheskoe predstavlenie dvizheniya v odnoi zadache o vzaimodeistvii tela so sredoi”, Prikl. mekh., 40:4 (2004), 137–144 | MR | Zbl

[156] Shamolin M. V., Nekotorye zadachi differentsialnoi i topologicheskoi diagnostiki, Ekzamen, M., 2004

[157] Shamolin M. V., “Ob odnom integriruemom sluchae v dinamike na $\mathrm{so}(4)\times\mathbf R^4$”, Tez. dokl. Vseross. konf. “Differentsialnye uravneniya i ikh prilozheniya” (SamDif-2005), Univers-grupp, Samara, 2005, 97–98

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

[159] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. Mezhdunar. konf. “Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz”, posv. 100-letiyu S. M. Nikolskogo (Moskva, 23–29 maya 2005 g.), Matem. in-t im. V. A. Steklova RAN, M., 2005, 244

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

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

[162] Shamolin M. V., “K zadache o prostranstvennom tormozhenii tvërdogo tela v soprotivlyayuscheisya srede”, Izv. RAN. MTT, 2006, no. 3, 45–57

[163] Shamolin M. V., Modelnaya zadacha o dvizhenii tela v soprotivlyayuscheisya srede s uchëtom zavisimosti momenta sily soprotivleniya ot uglovoi skorosti, Nauchnyi otchët In-ta mekhaniki MGU No 4818, M., 2006

[164] Shamolin M. V., “O sluchae polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 10–15 avgusta 2006 g.), Vladimir. gos. un-t, Vladimir, 2006, 226–228

[165] Shamolin M. V., “Integriruemost v elementarnykh funktsiyakh sistem s peremennoi dissipatsiei: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 38

[166] Shamolin M. V., “Integriruemost zadachi o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 21

[167] Shamolin M. V., “Integriruemost silno nekonservativnykh sistem v transtsendentnykh elementarnykh funktsiyakh: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 40

[168] Shamolin M. V., Metody analiza dinamicheskikh sistem s peremennoi dissipatsiei v dinamike tvërdogo tela, Ekzamen, M., 2007

[169] Shamolin M. V., Nekotorye zadachi differentsialnoi i topologicheskoi diagnostiki, Ekzamen, M., 2007

[170] Shamolin M. V., “Nekotorye modelnye zadachi dinamiki tvërdogo tela pri vzaimodeistvii ego so sredoi”, Prikl. mekh., 43:10 (2007), 49–67 | MR | Zbl

[171] Shamolin M. V., “Novye integriruemye sluchai v dinamike chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 27

[172] Shamolin M. V., “Ob integriruemosti v transtsendentnykh funktsiyakh: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 34

[173] Shamolin M. V., “Ob integriruemosti dvizheniya chetyrëkhmernogo tvërdogo tela-mayatnika, nakhodyaschegosya v ,potoke nabegayuschei sredy: Zasedaniya seminara mekhaniko-matematicheskogo fakulteta MGU im. M. V. Lomonosova ‘Aktualnye problemy geometrii i mekhaniki’ ”, SMFN, 23, 2007, 37

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

[175] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. Vseross. konf. “Sovremennye problemy mekhaniki sploshnoi sredy” pamyati L. I. Sedova v svyazi so 100-letiem so dnya rozhd. (Moskva, 12–14 noyabrya 2007 g.), MIAN, M., 2007, 166–167

[176] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole sil”, Tez. dokl. Mezhdunar. konf. “Analiz i osobennosti”, posvyasch. 70-letiyu V. I. Arnolda (Moskva), MIAN, M., 20–24 avgusta 2007 g., 110–112

[177] Shamolin M. V., “Sluchai polnoi integriruemosti v elementarnykh funktsiyakh nekotorykh klassov nekonservativnykh dinamicheskikh sistem”, Tez. dokl. Mezhdunar. konf. “Klassicheskie zadachi dinamiki tvërdogo tela” (09–13 iyunya 2007 g.), In-t prikl. matem. i mekhan. NAN Ukrainy, Donetsk, 2007, 81–82

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

[179] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole sil”, Nelineinyi dinamicheskii analiz-2007, Tez. dokl. mezhdunar. kongr. (Sankt-Peterburg, 4–8 iyunya 2007 g.), Sankt-Peterb. gos. un-t, SPb., 2007, 178

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

[181] Shamolin M. V., “Novye integriruemye sluchai v dinamike tela, vzaimodeistvuyuschego so sredoi, pri uchëte zavisimosti momenta sily soprotivleniya ot uglovoi skorosti”, Prikl. matematika i mekhanika, 72:2 (2008), 273–287 | MR | Zbl

[182] Shamolin M. V., “Novye sluchai polnoi integriruemosti v dinamike simmetrichnogo chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Mater. mezhdunar. nauch. konf. “Sovremennye problemy matematiki, mekhaniki, informatiki”, posv. 85-letiyu so dnya rozhd. L. A. Tolokonnikova (Tula, 17–21 noyabrya 2008 g.), Grif i K, Tula, 2008, 317–320

[183] Shamolin M. V., “Novyi integriruemyi sluchai v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole sil”, Mater. Voronezh. ves. matem. shk. “Pontryaginskie chteniya-XIX” (Voronezh, mai 2008 g.), Izd-vo Voronezh. gos. un-ta, Voronezh, 2008, 231–232

[184] 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 | MR | Zbl

[185] Shamolin M. V., “Trëkhparametricheskoe semeistvo fazovykh portretov v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Dokl. RAN, 418:1 (2008), 46–51 | MR | Zbl

[186] Shamolin M. V., “Klassifikatsiya sluchaev polnoi integriruemosti v dinamike simmetrichnogo chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Sovrem. mat. i eë pril., 65 (2009), 132–142

[187] Shamolin M. V., “Nekotorye sluchai polnoi integriruemosti v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, V Polyakhovskie chteniya, Tr. Mezhdunar. nauch. konf. po mekhan. (Sankt-Peterburg, 3–6 fevralya 2009 g.), SPbGU, Spb., 2009, 144–150

[188] Shamolin M. V., “Novye sluchai integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole sil: Tez. zased. sem. ‘Aktualnye problemy geometrii i mekhaniki’ ”, Sovrem. matem. i eë pril., 65 (2009), 6

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

[190] Shamolin M. V., “Novye sluchai polnoi integriruemosti v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. nauch. konf. “Lomonosovskie chteniya-2009”, sekts. mekhaniki (Moskva, MGU, aprel 2009 g.), Izd-vo Mosk. un-ta, M., 2009, 153–154

[191] Shamolin M. V., “Ob integriruemosti v elementarnykh funktsiyakh nekotorykh klassov nekonservativnykh dinamicheskikh sistem”, Sovrem. mat. i eë pril., 62 (2009), 131–171

[192] Shamolin M. V., “Ob ustoichivosti pryamolineinogo postupatelnogo dvizheniya”, Prikl. mekh., 45:6 (2009), 125–140 | MR | Zbl

[193] Shamolin M. V., “Sluchai integriruemosti uravnenii dvizheniya chetyrëkhmernogo tvërdogo tela v nekonservativnom pole sil”, Sovremennye problemy matematiki, mekhaniki i ikh prilozhenii, Mater. mezhdunar. konf., posvyasch. 70-letiyu rektora MGU akad. V. A. Sadovnichego (Moskva, 30 marta – 02 aprelya 2009 g.), Universitetskaya kniga, M., 2009, 233

[194] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike simmetrichnogo chetyrëkhmernogo tvërdogo tela v nekonservativnom pole: Tez. zased. sem. ‘Aktualnye problemy geometrii i mekhaniki’ ”, Sovrem. matem. i eë pril., 65 (2009), 9

[195] Shamolin M. V., “Integriruemost i neintegriruemost v transtsendentnykh funktsiyakh dinamicheskikh sistem”, Voronezhskaya zimnyaya matem. shk. S. G. Kreina 2010, Tez. dokl., VGU, Voronezh, 2010, 159–160

[196] Shamolin M. V., K zadache o dvizhenii tela s perednim ploskim tortsom v soprotivlyayuscheisya srede, Nauchnyi otchët In-ta mekhaniki MGU No 5052, In-t mekhaniki MGU, M., 2010

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

[198] Shamolin M. V., “Prostranstvennoe dvizhenie tvërdogo tela v srede s soprotivleniem”, Prikl. mekh., 46:7 (2010), 120–133 | MR

[199] Shamolin M. V., “Sluchai polnoi integriruemosti uravnenii dvizheniya dinamicheski simmetrichnogo chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 2–7 iyulya 2010 g.), Vladimir. gos. un-t, Vladimir, 2010, 195

[200] Shamolin M. V., “Sluchai polnoi integriruemosti uravnenii prostranstvennogo dvizheniya tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. XI Mezhdunar. konf. “Ustoichivost i kolebaniya nelineinykh sistem upravleniya” (Moskva, IPU RAN, 1–4 iyunya 2010 g.), IPU RAN, M., 2010, 429–431

[201] Shamolin M. V., “Sluchai polnoi integriruemosti uravnenii prostranstvennoi dinamiki tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. nauch. konf. “Lomonosovskie chteniya-2010”, sekts. mekhaniki (Moskva, MGU, aprel 2010 g.), Izd-vo Mosk. un-ta, M., 2010, 172

[202] Shamolin M. V., “Sluchai polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, UMN, 65:1 (2010), 189–190 | DOI | MR | Zbl

[203] Shamolin M. V., “Dvizhenie tvërdogo tela v soprotivlyayuscheisya srede”, Matem. modelirovanie, 23:12 (2011), 79–104 | MR | Zbl

[204] Shamolin M. V., “Dinamicheskie invarianty integriruemykh dinamicheskikh sistem s peremennoi dissipatsiei”, Vestn. Nizhegorod. un-ta im. N. I. Lobachevskogo, 2011, no. 4, 356–357

[205] Shamolin M. V., “Mnogoparametricheskoe semeistvo fazovykh portretov v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2011, no. 3, 24–30 | MR

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

[207] Shamolin M. V., “Novyi sluchai polnoi integriruemosti uravnenii dinamiki na kasatelnom rassloenii k trëkhmernoi sfere”, Vestn. SamGU. Estestvennonauch. ser., 2011, no. 5(86), 187–189

[208] Shamolin M. V., “Polnye spiski pervykh integralov v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Mezhdunar. konf., posvyasch. 110-i godovschine I. G. Petrovskogo, XXIII sovmestn. zased. MMO i sem. im. I. G. Petrovskogo. Tez. dokl., Izd-vo Mosk. un-ta, Intuit.RU, M., 2011, 389–390

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

[210] Shamolin M. V., “Zadacha o dvizhenii tela v soprotivlyayuscheisya srede s uchëtom zavisimosti momenta sily soprotivleniya ot uglovoi skorosti”, Matem. modelirovanie, 24:10 (2012), 109–132 | MR

[211] Shamolin M. V., “Nekotorye voprosy kachestvennoi teorii v dinamike sistem s peremennoi dissipatsiei”, Sovrem. mat. i eë pril., 78 (2012), 138–147

[212] Shamolin M. V., “Novye sluchai integriruemosti v transtsendentnykh funktsiyakh v dinamike tvërdogo tela v nekonservativnom pole”, Sovremennye metody teorii kraevykh zadach, Mater. Voronezh. vesennei matem. shk. “Pontryaginskie chteniya-XXIII” (Voronezh, 3–9 maya 2012 g.), Voronezh. gos. un-t, Voronezh, 2012, 200

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

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

[215] Shamolin M. V., “Novyi sluchai integriruemosti v transtsendentnykh funktsiyakh v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, XII Mezhdunar. konf. “Ustoichivost i kolebaniya nelineinykh sistem upravleniya”, Konf. Pyatnitskogo (Moskva, 5–8 iyunya 2012 g.), Tez. dokl., IPU RAN, M., 2012, 339–341

[216] Shamolin M. V., “Obzor sluchaev integriruemosti v dinamike malomernogo i mnogomernogo tvërdogo tela v nekonservativnom pole”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 29 iyunya – 4 iyulya 2012 g.), Suzdal, 2012, 179–180

[217] Shamolin M. V., “Polnyi spisok pervykh integralov dinamicheskikh uravnenii dvizheniya tvërdogo tela v soprotivlyayuscheisya srede pri uchëte lineinogo dempfirovaniya”, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2012, no. 4, 44–47 | MR | Zbl

[218] Shamolin M. V., “Sluchai integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Voronezhskaya zimn. matem. shk. S. G. Kreina 2012, Mater. mezhdunar. konf. (Voronezh, 25–30 yanvarya 2012 g.), Izd-vo VGU, 2012, 213–215

[219] Shamolin M. V., “Sluchai integriruemosti v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi pri struinom obtekanii”, VI Polyakhovskie chteniya, Tez. dokl. Mezhdunar. nauchn. konf. po mekhan. (Sankt-Peterburg, 31 yanvarya – 3 fevralya 2012 g.), I. V. Balabanov, M., 2012, 75

[220] Shamolin M. V., “Sluchai integriruemosti v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi, pri uchëte lineinogo dempfirovaniya”, Analiticheskaya mekhanika, ustoichivost i upravlenie, Tr. X Mezhdunar. Chetaevskoi konf. (Kazan, 12–16 iyunya 2012 g.), v. 1, Sektsiya 1. Analiticheskaya mekhanika, Izd-vo Kazan. gos. tekhn. un-ta, Kazan, 2012, 508–514

[221] Shamolin M. V., “Sluchai polnoi integriruemosti v transtsendentnykh funktsiyakh v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi: Tez. zased. sem. ‘Aktualnye problemy geometrii i mekhaniki’ ”, Sovrem. matem. i eë pril., 76 (2012), 7

[222] Shamolin M. V., “Sopostavlenie sluchaev polnoi integriruemosti v dinamike dvumernogo, trëkhmernogo i chetyrëkhmernogo tvërdogo tela v nekonservativnom pole”, Sovrem. mat. i eë pril., 76 (2012), 84–99

[223] Shamolin M. V., “Mnogoobrazie sluchaev integriruemosti v dinamike malomernogo i mnogomernogo tvërdogo tela v nekonservativnom pole sil”, Itogi nauki i tekhn. Ser. Sovrem. mat. i eë pril. Temat. obzory, 125, 2013, 5–254

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

[225] Shamolin M. V., “Novyi sluchai integriruemosti transtsendentnykh funktsiyakh v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Avtomatika i telemekhanika, 2013, no. 8, 173–190 | MR | Zbl

[226] Shamolin M. V., “Novyi sluchai integriruemosti uravnenii dinamiki na kasatelnom rassloenii k trëkhmernoi sfere”, UMN, 68:5 (2013), 185–186 | DOI | MR | Zbl

[227] Shamolin M. V., “Ob integriruemosti v zadachakh dinamiki tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Prikl. mekhan., 49:6 (2013), 44–54 | MR | Zbl

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

[229] Shamolin M. V., “Zadacha o dvizhenii tela v soprotivlyayuscheisya srede pod deistviem sledyaschei sily: kachestvennyi analiz i integriruemost”, XII Vseross. sovesch. po problemam upravleniya, VSPU-2014 (Moskva, 16–19 iyunya 2014 g.), Trudy [Elektronnyi resurs], IPU RAN, M., 2014, 1813–1824

[230] Shamolin M. V., “Integriruemye sistemy s peremennoi dissipatsiei na kasatelnom rassloenii k konechnomernoi sfere”, Geom. analiz i ego pril., Mater. II Mezhdunar. konf. (Volgograd, 26–30 maya 2014 g.), Izd-vo VolGU, Volgograd, 2014, 143–145

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

[232] Shamolin M. V., “Obzor sluchaev integriruemosti uravnenii dvizheniya mnogomernogo tvërdogo tela v nekonservativnom pole”, Mater. mezhdunar. konf. Voronezhskaya zimnyaya matem. shk. S. G. Kreina–2014, ed. V. A. Kostin, Nauchnaya kniga, Voronezh, 2014, 404–408

[233] Shamolin M. V., “Integriruemye sistemy na kasatelnom rassloenii k mnogomernoi sfere”, Tr. semin. im. I. G. Petrovskogo (to appear)

[234] Shamolin M. V., Tsyptsyn S. V., Analiticheskoe i chislennoe issledovanie traektorii dvizheniya tela v soprotivlyayuscheisya srede, Nauchnyi otchët In-ta mekhaniki MGU No 4289, M., 1993

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

[237] Peixoto M., “On structural stability”, Ann. Math. (2), 69 (1959), 199–222 | DOI | MR | Zbl

[238] Peixoto M., “Structural stability on two-dimensional manifolds”, Topology, 1:2 (1962), 101–120 | DOI | MR | Zbl

[239] Peixoto M., “On an approximation theorem of Kupka and Smale”, J. Differ. Equ., 3 (1966), 214–227 | DOI | MR

[240] Prandtl L., Betz A., Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen, R. Oldenbourg, Berlin, 1932

[241] Shamolin M. V., “Structural stability in 3D dynamics of a rigid body”, CD Proc. of WCSMO-3 (Buffalo, NY, May 17–21, 1999), Buffalo, NY, 1999

[242] Shamolin M. V., “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability”, CD Proc. of ECCOMAS 2000 (Barcelona, Spane, 11–14 September, 2000), Barcelona, 2000

[243] 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), 2526–2555 | DOI | MR

[244] Shamolin M. V., “Foundations of differential and topological diagnostics”, J. Math. Sci., 114:1 (2003), 976–1024 | DOI | MR | Zbl

[245] Shamolin M. V., “New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium”, J. Math. Sci., 114:1 (2003), 919–975 | DOI | MR | Zbl

[246] Shamolin M. V., “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body”, J. Math. Sci., 122:1 (2004), 2841–2915 | DOI | MR | Zbl

[247] Shamolin M. V., “Some cases of integrability in dynamics of a rigid body interacting with a resisting medium”, Tez. dokl. Mezhdunar. konf. po differents. uravn. i dinam. sist. (Suzdal, 5–10 iyulya 2004 g.), Vladimir. gos. un-t, Vladimir, 2004, 296–298

[248] Shamolin M. V., “Some cases of integrability in 3D dynamics of a rigid body interacting with a medium”, Book of Abstracts IMA Int. Conf. “Recent Advances in Nonlinear Mechanics” (Aberdeen, Scotland, August 30 – September 1, 2005), IMA, Aberdeen, 2005, 112

[249] Shamolin M. V., “Structural stable vector fields in rigid body dynamics”, Proc. of 8th Conf. on Dynamical Systems (Theory and Applications), DSTA 2005 (Lodz, Poland, Dec. 12–15, 2005), v. 1, Tech. Univ. Lodz, 2005, 429–436

[250] Shamolin M. V., “The cases of complete integrability in dynamics of a rigid body interacting with a medium”, Book of Abs. of Int. Conf. on the Occasion of the 150th Birthday of A. M. Lyapunov (June 24–30, 2007, Kharkiv, Ukraine), Verkin Inst. Low Temper. Phys. Eng., Kharkiv, 2007, 147–148

[251] Shamolin M. V., “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium”, Proc. of 9th Conf. on Dynamical Systems (Theory and Applications), DSTA 2007 (Lodz, Poland, Dec. 17–20, 2007), v. 1, Tech. Univ. Lodz, 2007, 415–422

[252] Shamolin M. V., “4D rigid body and some cases of integrability”, Abstracts of ICIAM07 (Zurich, Switzerland, June 16–20, 2007), ETH Zurich, 2007, 311

[253] Shamolin M. V., “Methods of analysis of dynamic systems with various dissipation in dynamics of a rigid body”, ENOC-2008, CD Proc. (June 30 – July 4, 2008, Saint Petersburg, Russia)

[254] Shamolin M. V., “Some methods of analysis of the dynamic systems with various dissipation in dynamics of a rigid body”, Proc. Appl. Math. Mech., 8 (2008), 10137–10138 | DOI

[255] Shamolin M. V., “Dynamical systems with variable dissipation: methods and applications”, Proc. of 10th Conf. on Dynamical Systems (Theory and Applications), DSTA 2009 (Lodz, Poland, Dec. 7–10, 2009), Tech. Univ. Lodz, 2009, 91–104

[256] Shamolin M. V., “New cases of integrability in dynamics of a rigid body with the cone form of its shape interacting with a medium”, Proc. Appl. Math. Mech., 9 (2009), 139–140 | DOI

[257] Shamolin M. V., “The various cases of complete integrability in dynamics of a rigid body interacting with a medium”, Multibody Dynamics, ECCOMAS Thematic Conf. (Warsaw, Poland, 29 June – 2 July 2009), Book of Abstracts, Polish Acad. Sci., Warsaw, 2009, 276–277

[258] Shamolin M. V., “The various cases of complete integrability in dynamics of a rigid body interacting with a medium”, Multibody Dynamics, ECCOMAS Thematic Conf. (Warsaw, Poland, 29 June – 2 July 2009), CD Proc., Polish Acad. Sci., Warsaw, 2009

[259] Shamolin M. V.,, “Dynamical systems with various dissipation: background, methods, applications”, CD Proc. of XXXVIII Summer School-Conf. “Advances Problems in Mechanics”, APM 2010 (July 1–5, 2010, St. Petersburg (Repino), Russia), IPME, St. Petersburg, 2010, 612–621

[260] Shamolin M. V., “Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid body”, Proc. Appl. Math. Mech., 10 (2010), 63–64 | DOI

[261] Shamolin M. V., “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices”, CD Proc. 5th Int. Sci. Conf. on Physics and Control PHYSCON 2011 (Leon, Spain, September 5–8, 2011), Leon, 2011

[262] Shamolin M. V., “Variety of the cases of integrability in dynamics of a 2D-, 3D-, and 4D-rigid body interacting with a medium”, Proc. of 11th Conf. on Dynamical Systems (Theory and Applications), DSTA 2011 (Lodz, Poland, Dec. 5–8, 2011), v. 1, Tech. Univ. Lodz, 2011, 11–24

[263] Shamolin M. V., “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices”, 83rd Ann. Sci. Conf. of the Int. Assoc. of Appl. Math. and Mech. (Darmstadt, Germany, March 26–30, 2012), Book of Abstracts, TU Darmstadt, Darmstadt, 2012

[264] Shamolin M. V., “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices”, Proc. Appl. Math. Mech., 12 (2012), 43–44 | DOI

[265] Shamolin M. V., “Cases of integrability in dynamics of a rigid body interacting with a resistant medium”, Abstract Book. 23th Int. Congress of Theor. and Appl. Mech. (August 19–24, 2012, Beijing, China), Sci. Lit. Publ. House, Beijing, 2012, 51

[266] Shamolin M. V., “Cases of integrability in dynamics of a rigid body interacting with a resistant medium”, CD proc. 23th Int. Congress of Theor. and Appl. Mech. (August 19–24, 2012, Beijing, China), Sci. Lit. Publ. House, Beijing, 2012

[267] Shamolin M. V., “Variety of the cases of integrability in dynamics of a 2D-, and 3D-rigid body interacting with a medium”, 8th ESMC 2012 (Graz, Austria, July 9–13, 2012), CD Materials, Graz, 2012

[268] Shamolin M. V., “Cases of integrability in transcendental functions in 3D Dynamics of a rigid body interacting with a medium”, Proc. ECCOMAS Multibody Dynamics 2013 (1–4 July, 2013, Univ. of Zagreb, Croatia), Univ. of Zagreb, Zagreb, 2013, 903–912

[269] Shamolin M. V., “Dynamical pendulum-like nonconservative systems”, Proc. of 12th Conf. on Dynamical Systems (Theory and Applications), DSTA 2013 (Lodz, Poland, Dec. 2–5, 2013), Tech. Univ. Lodz, 2013, 160

[270] Shamolin M. V., “Review of cases of integrability in dynamics of low- and multidimensional rigid body in a nonconservative field”, XXXIII Int. Conf. Dynamics Days Europe 2013 (3–7 June 2013, Madrid, Spain), Book of Abstracts, CTB UPM, Madrid, 2013, 157 http://www.dynamics-days-europe-2013.org/DDEXXXIII-AbstractsBook.pdf

[271] Shamolin M. V., “Review of integrable cases in dynamics of low- and multidimensional rigid body in a nonconservative field”, Advanced Problems in Mechanics, Book of Abstracts of Int. Summer School-Conf. (1–6 of July 2013, St. Petersburg), Polytech. Univ. Publ. House, St. Petersburg, 2013, 99

[272] Shamolin M. V., “Variety of the cases of integrability in dynamics of a symmetric 2D-, 3D-, and 4D-rigid body in a nonconservative field”, Int. J. Struct. Stabil. Dynam., 13:7 (2013), 1340011 | DOI | MR

[273] Shamolin M. V., “Review of cases of integrability in dynamics of lower- and multidimensional rigid body in a nonconservative field of forces”, Recent Advances in Mathematics, Statistics and Economics, Proc. of 2014 Int. EUROPMENT Conf. on Pure Math. Appl. Math., PM–AM'14 (Venice, Italy, March 15–17, 2014), Venice, 2014, 86–102

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