Geodesic
Dynamical systems with variable dissipation: Approaches, methods, and applications
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 3, pp. 3-237.

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This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can call the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under the streamline flow around conditions. The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense, i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci). The author obtains new families of phase portraits of systems with variable dissipation on lower- and higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness. He discovers new integrable cases of the rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in the over-running medium flow. 
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M. V. Shamolin. Dynamical systems with variable dissipation: Approaches, methods, and applications. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 3, pp. 3-237. http://geodesic.mathdoc.fr/item/FPM_2008_14_3_a0/

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[283] 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

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[302] Shamolin M. V., “Nekotorye zadachi prostranstvennoi dinamiki tvërdogo tela, vzaimodeistvuyuschego so sredoi v usloviyakh kvazistatsionarnosti”, Tez. dokl. Vseros. nauchn.-tekhn. konf. molodykh uchënykh “Sovremennye problemy aerokosmicheskoi nauki” (Zhukovskii, 27–29 maya 1998 g.), Izd-vo TsAGI, M., 1998, 89–90

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

[304] Shamolin M. V., “Semeistva trëkhmernykh fazovykh portretov v prostranstvennoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, III Mezhdunar. simp. po klassicheskoi i nebesnoi mekhanike, Tez. dokl. (Velikie Luki, 23–27 avgusta 1998 g.), VTs RAN, M.–Velikie Luki, 1998, 165–167

[305] Shamolin M. V., “Semeistvo portretov s predelnymi tsiklami v ploskoi dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Izv. RAN. MTT, 1998, no. 6, 29–37 | MR

[306] Shamolin M. V., “Metody nelineinogo analiza v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi (Methods of nonlinear analysis in dynamics of a rigid interacting with a medium)”, CD Proc. of the Cong. “Nonlinear Analysis and Its Applications” (Moscow, Russia, September 1–5, 1998), 1999, 497–508

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

[308] Shamolin M. V., “Nelineinye dinamicheskie effekty pri prostranstvennom tormozhenii tela v soprotivlyayuscheisya srede”, Tez. dokl. III Mezhdunar. konf. “Chkalovskie chteniya. Inzhenerno-fizicheskie problemy aviatsii i kosmicheskoi tekhniki” (1–4 iyunya 1999 g.), EATK GA, Egorevsk, 1999, 257–258

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

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

[311] Shamolin M. V., “Semeistva dlinnoperiodicheskikh traektorii v prostranstvennoi dinamike tvërdogo tela”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Tez. dokl. (System Modelling). Nauchn. konf. (25–29 maya 1999 g.), Kiev, 1999, 60

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

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

[315] Shamolin M. V., “Mnogomernye topograficheskie sistemy Puankare i transtsendentnaya integriruemost”, IV Sibirskii kongr. po prikladnoi i industrialnoi matematike, Tez. dokl. Ch. I (Novosibirsk, 26 iyunya – 1 iyulya 2000 g.), Izd-vo In-ta matematiki, Novosibirsk, 2000, 25–26

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

[317] Shamolin M. V., “O grubosti dissipativnykh sistem i otnositelnoi grubosti sistem s peremennoi dissipatsiei”, Tez. dokl. seminara po vektornomu i tenzornomu analizu im. P. K. Rashevskogo, Vestn. Mosk. un-ta. Ser. 1. Matematika, mekhanika, 2000, no. 2, 63

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

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

[320] Shamolin M. V., “Sopostavlenie nekotorykh integriruemykh sluchaev iz dvumernoi, trëkhmernoi i chetyrëkhmernoi dinamiki tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. dokl. V Krymskoi mezhdunar. mat. shkoly “Metod funktsii Lyapunova i ego prilozheniya”, MFL–2000 (Krym, Alushta, 5–13 sentyabrya 2000 g.), Simferopol, 2000, 169

[321] Shamolin M. V., “Zadacha diagnostirovaniya kak glavnaya zadacha obschei zadachi differentsialnoi diagnostiki”, Book of Abstracts of the Third Int. Conf. “Tools for Mathematical Modelling” (Saint-Petersburg, Russia, 18–23 June, 2001), Saint-Petersburg State Tech. Univ., Saint-Petersburg, 2001, 121

[322] Shamolin M. V., “Integriruemost zadachi o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Fundament. i prikl. mat., 7:1 (2001), 309

[323] Shamolin M. V., “Mnogoobrazie tipov fazovykh portretov v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Fundament. i prikl. mat., 7:1 (2001), 302–303

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

[325] 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), Nauchn. konf. Thes. of Conf. Rep. (22–25 maya 2001 g.), Kyiv, 2001, 344

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

[327] 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

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

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

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

[331] Shamolin M. V., “Integriruemost v transtsendentnykh funktsiyakh v dinamike tvërdogo tela”, Tez. dokl. nauchn. konf. “Lomonosovskie chteniya–2003”, Sektsiya mekhaniki (Moskva, MGU im. M. V. Lomonosova, 17–27 aprelya 2003 g.), Izd-vo Mosk. un-ta, M., 2003, 130

[332] Shamolin M. V., “Ob integriruemosti nekonservativnykh dinamicheskikh sistem v transtsendentnykh funktsiyakh”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Nauchn. konf. Thes. of Conf. Rep. (27–30 maya 2003 g.), Kyiv, 2003, 377

[333] Shamolin M. V., “Ob odnoi prostranstvennoi zadache o dvizhenii tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. Mezhdunar. nauchn. konf. po mekhanike “Treti Polyakhovskie chteniya” (Sankt-Peterburg, 4–6 fevralya 2003 g.), Izd-vo NIIKh Sankt-Peterburgskogo un-ta, SPb., 2003, 170–171

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

[335] Shamolin M. V., “Integriruemost nekonservativnykh sistem v transtsendentnykh elementarnykh funktsiyakh”, Mat. konf. “X Mizhd. nauk. konf. im. akad. M. Kravchuka” (Kiev, 13–15 tr. 2004), Kiev, 2004, 279

[336] Shamolin M. V., Metody analiza klassov nekonservativnykh sistem v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi, Dis. $\dots$ dokt. fiz.-mat. nauk, MGU, M., 2004

[337] Shamolin M. V., Metody analiza klassov nekonservativnykh sistem v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi, Avtoreferat dis. $\dots$ dokt. fiz.-mat. nauk, M., 2004

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

[339] Shamolin M. V., “Dinamicheskie sistemy s peremennoi dissipatsiei v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Differentsialnye uravneniya i sistemy kompyuternoi algebry (Differential equations and computer algebra systems, DE CAS' 2005), Mater. Mezhdunar. konf. Ch. 1 (Brest, 5–8 oktyabrya 2005 g.), Izd-vo BGPU, Minsk, 2005, 231–233

[340] Shamolin M. V., “Integriruemost v transtsendentnykh funktsiyakh v dinamike tvërdogo tela”, Mat. konf. “Sovremennye problemy prikladnoi matematiki i matematicheskogo modelirovaniya” (Voronezh, 12–17 dekabrya 2005 g.), Voronezhskaya gos. akademiya, Voronezh, 2005, 240

[341] Shamolin M. V., “O dvizhenii tvërdogo tela v soprotivlyayuscheisya srede pri uchëte vraschatelnykh proizvodnykh momenta aerodinamicheskikh sil po uglovoi skorosti”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Nauchn. konf. Thes. of. Conf. Rep. (23–25 maya 2005 g.), Kyiv, 2005, 351

[342] Shamolin M. V., “O dvizhenii tela v soprotivlyayuscheisya srede pri uchëte vraschatelnykh proizvodnykh momenta aerodinamicheskikh sil po uglovoi skorosti”, Tez. dokl. nauchn. konf. “Lomonosovskie chteniya–2005”, Sektsiya mekhaniki (Moskva, MGU im. M. V. Lomonosova, Aprel 2005 g.), Izd-vo Mosk. un-ta, M., 2005, 182

[343] Shamolin M. V., “Ob odnom integriruemom sluchae v dinamike na $\operatorname{so}(4)\times\mathbb R^4$”, Tez. dokl. Vseros. konf. “Differentsialnye uravneniya i ikh prilozheniya”, SamDif–2005 (Samara, 27 iyunya – 2 iyulya 2005 g.), Univers-grupp, Samara, 2005, 97–98

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

[345] 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.), MIAN im. V. A. Steklova, M., 2005, 244

[346] 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

[347] 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

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

[349] Shamolin M. V., “K prostranstvennoi zadache vzaimodeistviya tvërdogo tela s soprotivlyayuscheisya sredoi”, Ann. dokl. IX Vseros. s'ezda po teoreticheskoi i prikladnoi mekhanike, T. I (Nizhnii Novgorod, 22–28 avgusta 2006 g.), Izd-vo Nizhegorodskogo gos. un-ta im. N. I. Lobachevskogo, Nizhnii Novgorod, 2006, 120

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

[351] Shamolin M. V., “O sluchae polnoi integriruemosti v dinamike chetyrëkhmernogo tvërdogo tela”, Tez. dokl. Mezhdunar. konf. po differentsialnym uravneniyam i dinamicheskim sistemam (Vladimir, 10–15 iyulya 2006 g.), Vladimirskii gos. un-t, Vladimir, 2006, 226–228

[352] Shamolin M. V., “O traektoriyakh kharakternykh tochek tvërdogo tela, dvizhuschegosya v srede”, Mezhdunar. konf. “Pyatye Okunevskie chteniya”, Tez. dokl. (Sankt-Peterburg, 26–30 iyunya 2006 g.), Baltiiskii gos. tekhn. un-t, SPb., 2006, 34

[353] Shamolin M. V., “Prostranstvennaya zadacha o dvizhenii tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. VIII Krymskoi mezhdunar. mat. shkoly “Metod funktsii Lyapunova i ego prilozheniya” (Alushta, Tavricheskii nats. un-t, 10–17 sentyabrya 2006 g.), DiAiPi, Simferopol, 2006, 184

[354] Shamolin M. V., “Sistemy s peremennoi dissipatsiei v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Chetvërtye Polyakhovskie chteniya, Tez. dokl. Mezhdunar. nauchn. konf. po mekhanike (Sankt-Peterburg, 7–10 fevralya 2006 g.), VVM, SPb., 2006, 86

[355] Shamolin M. V., “Vliyanie vraschatelnykh proizvodnykh momenta sily vozdeistviya sredy po uglovoi skorosti tvërdogo tela na ego dvizhenie”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 44 | MR

[356] Shamolin M. V., “Integriruemost v elementarnykh funktsiyakh sistem s peremennoi dissipatsiei”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 38

[357] Shamolin M. V., “Integriruemost zadachi o dvizhenii chetyrëkhmernogo tvërdogo tela v soprotivlyayuscheisya srede”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 21

[358] Shamolin M. V., “Integriruemost silno nekonservativnykh sistem v transtsendentnykh elementarnykh funktsiyakh”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 40

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

[360] Shamolin M. V., “Mnogoobrazie tipov fazovykh portretov v dinamike tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 17

[361] Shamolin M. V., Nekotorye zadachi differentsialnoi i topologicheskoi diagnostiki, 2-e izd., Ekzamen, M., 2007

[362] Shamolin M. V., “Novye integriruemye sluchai v dinamike chetyrëkhmernogo tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 27

[363] Shamolin M. V., “O dvizhenii tvërdogo tela v soprotivlyayuscheisya srede pri uchëte vraschatelnykh proizvodnykh momenta aerodinamicheskikh sil po uglovoi skorosti”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 44 | MR

[364] Shamolin M. V., “O rabote Vserossiiskoi konferentsii “Differentsialnye uravneniya i ikh prilozheniya”, Samara, 27 iyunya – 2 iyulya 2005 g.”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 45

[365] Shamolin M. V., “Ob integriruemosti v transtsendentnykh funktsiyakh”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 34

[366] Shamolin M. V., “Ob integriruemosti v elementarnykh funktsiyakh nekotorykh klassov nekonservativnykh dinamicheskikh sistem”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Int. Conf. Thes. of Conf. Rep. (22–25 May 2007), Kyiv, 2007, 249

[367] Shamolin M. V., “Ob integriruemosti dvizheniya chetyrëkhmernogo tvërdogo tela-mayatnika, nakhodyaschegosya v potoke nabegayuschei sredy”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 37

[368] Shamolin M. V., “Ob ustoichivosti odnogo rezhima dvizheniya tvërdogo tela v soprotivlyayuscheisya srede”, Tez. dokl. nauchn. konf. “Lomonosovskie chteniya–2007”, Sektsiya mekhaniki (Moskva, MGU im. M. V. Lomonosova, Aprel 2007 g.), Izd-vo Mosk. un-ta, M., 2007, 153

[369] Shamolin M. V., “Ob uchëte vraschatelnykh proizvodnykh momenta aerodinamicheskikh sil na dvizhenie tela v soprotivlyayuscheisya srede”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 26

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

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

[372] 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, 20–24 avgusta 2007 g.), MIAN im. V. A. Steklova, M., 2007, 110–112

[373] Shamolin M. V., “Sluchai polnoi integriruemosti v elementarnykh funktsiyakh nekotorykh klassov nekonservativnykh dinamicheskikh sistem”, Tez. dokl. Mezhdunar. konf. “Klassicheskie zadachi dinamiki tvërdogo tela” (9–13 iyunya 2007 g.), In-t prikl. mat. i mekhan. NAN Ukrainy, Donetsk, 2007, 81–82

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

[375] 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-Peterburgskii gos. un-t, SPb., 2007, 178

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

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

[378] Shamolin M. V., Shebarshov D. V., “Proektsii lagranzhevykh torov dlya bigarmonicheskogo ostsillyatora na prostranstvo polozhenii i dinamika tvërdogo tela, vzaimodeistvuyuschego so sredoi”, Modelirovanie i issledovanie ustoichivosti sistem (Modelling and Investigation of System Stability), Tez. dokl. (Mechanical Systems). Nauchn. konf. (19–23 maya 1997 g.), Kiev, 1997, 142

[379] Shamolin M. V., Shebarshov D. V., Metody resheniya osnovnoi zadachi differentsialnoi diagnostiki, Dep. v VINITI 12.05.99, No 1500-V99, M., 1999

[380] Shamolin M. V., Shebarshov D. V., Nekotorye voprosy geometrii v klassicheskoi mekhanike, Dep. v VINITI 12.05.99, No 1499-V99, M., 1999

[381] Shamolin M. V., Shebarshov D. V., “Nekotorye zadachi differentsialnoi diagnostiki”, Modelirovanie i issledovanie ustoichivosti sistem (Dynamical Systems Modelling and Stability Investigation), Tez. dokl. (System Modelling). Nauchn. konf. (25–29 maya 1999 g.), Kiev, 1999, 61

[382] Shamolin M. V., Shebarshov D. V., “Nekotorye zadachi differentsialnoi diagnostiki”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Fundament. i prikl. mat., 7:1 (2001), 305 | MR

[383] Shamolin M. V., Shebarshov D. V., “Nekotorye zadachi differentsialnoi diagnostiki”, Tez. zased. sem. “Aktualnye problemy geometrii i mekhaniki”, Sovremennaya matematika. Fundamentalnye napravleniya, 23, RUDN, M., 2007, 19

[384] Shorygin O. P., Shulman N. A., “Vkhod diska v vodu s uglom ataki”, Uch. zapiski TsAGI, 8:1 (1977), 12–21

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

[387] Yakobson M. V., “O gladkikh otobrazheniyakh okruzhnosti v sebya”, Mat. sb., 85(127):2 (1971), 163–188 | MR | Zbl

[388] Airy H., “The soaring of birds”, Nature, 28:709 (1883), 103 | DOI

[389] Blix M., “Une nouvelle théorie sur le vol a viole des oiseaux”, Revue generale sciences pures et appliquees, 1890

[390] Ishlinsky A. Yu., Klimov D. M., “Some aspects of the solution of the main problem of inertial navigation”, J. Inst. Navig., 23:4 (1970), 375–385

[391] Liliental O., Der Vogelflug als Grundlage der Fliegekunst, Berlin, 1889

[392] Marey E.-J., Le vol des oiseaux, Chap. XX, Paris, 1890

[393] Mouillard L.-P., L'empire de l'air, Paris, 1881

[394] Onniere B., “Étude sur le vol plane”, L'Aeronaute, 1891

[395] Von Parseval A., Die Mechanik des Vogelflugs, Wisbaden, 1889

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

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

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

[399] Peal S. E., “Soaring of birds”, Nature, 23 (1880), 10–11 | DOI

[400] Prandtl L., Betz A., Ergebnisse der aerodynamischen Versuchsanstalt zu Göttingen, Berlin, 1932

[401] Rayleigh (Strutt J. W.), “The soaring of birds”, Nature, 27:701 (1883), 534–535

[402] Shamolin M. V., “Global qualitative analysis of the nonlinear systems on the problem of a body motion in a resisting medium”, Fourth Colloquium on the Qualitative Theory of Differential Equations (Bolyai Institute, August 18–21, 1993), Szeged, Hungary, 1993, 54

[403] Shamolin M. V., “Relative structural stability on the problem of a body motion in a resisting medium”, ICM' 94, Abstract of Short Communications (Zurich, 3–11 August, 1994), Zurich, Switzerland, 1994, 207

[404] Shamolin M. V., “New two-parameter families of the phase patterns on the problem of a body motion in a resisting medium”, ICIAM' 95, Book of Abstracts (Hamburg, 3–7 July 1995), Hamburg, Germany, 1995, 436

[405] Shamolin M. V., “Poisson-stable and dense orbits in rigid body dynamics”, 3rd Experimental Chaos Conf., Advance Program (Edinburgh, Scotland, August 21–23, 1995), Edinburgh, Scotland, 1995, 114

[406] Shamolin M. V., “Qualitative methods to the dynamic model of an interaction of a rigid body with a resisting medium and new two-parametric families of the phase portraits”, DynDays' 95 (Sixteenth Annual Informal Workshop), Program and Abstracts (Lyon, June 28 – July 1, 1995), Lyon, France, 1995, 185

[407] Shamolin M. V., “Structural optimization of the controlled rigid motion in a resisting medium”, WCSMO–1, Extended Abstracts. Posters (Goslar, May 28 – June 2, 1995), Goslar, Germany, 1995, 18–19 | MR

[408] Shamolin M. V., “Qualitative methods in interacting with the medium rigid body dynamics”, Abstracts of GAMM Wissenschaftliche Jahrestagung' 96 (Karls-Universität Prague, Prague, Czech Rep., 27–31 May 1996), 129–130

[409] Shamolin M. V., “Qualitative methods in interacting with the medium rigid body dynamics”, Abstracts of XIXth ICTAM (Kyoto, Japan, August 25–31, 1996), Kyoto, Japan, 1996, 285

[410] Shamolin M. V., “Relative structural stability and relative structural instability of different degrees in topological dynamics”, Abstracts of Int. Topological Conf. Dedicated to P. S. Alexandroff's 100th Birthday “Topology and Applications” (Moscow, May 27–31, 1996), Phasys, Moscow, 1996, 207–208

[411] Shamolin M. V., “Topographical Poincaré systems in many dimensional spaces”, Fifth Colloquium on the Qualitative Theory of Differential Equations (Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, July 29 – August 2, 1996), Szeged, Hungary, 1996, 45

[412] Shamolin M. V., “Classical problem of a three-dimensional motion of a pendulum in a jet flow”, 3rd EUROMECH Solid Mechanics Conf., Book of Abstracts (Stockholm, Sweden, August 18–22, 1997), Royal Inst. of Technology, Stockholm, Sweden, 1997, 204

[413] Shamolin M. V., “Families of three-dimensional phase portraits in dynamics of a rigid body”, EQUADIFF 9, Abstracts, Enlarged Abstracts (Brno, Czech Rep., August 25–29, 1997), Masaryk Univ., Brno, Czech Rep., 1997, 76

[414] Shamolin M. V., “Three-dimensional structural optimization of controlled rigid motion in a resisting medium”, Proc. of WCSMO–2 (Zakopane, Poland, May 26–30, 1997), Zakopane, Poland, 1997, 387–392

[415] Shamolin M. V., “Three-dimensional structural optimization of controlled rigid motion in a resisting medium”, WCSMO–2, Extended Abstracts (Zakopane, Poland, May 26–30, 1997), Zakopane, Poland, 1997, 276–277

[416] Shamolin M. V., “Lyapunov functions method and many-dimensional topographical systems of Poincare in rigid body dynamics”, Tez. dokl. IV Krymskoi mezhdunar. mat. shkoly “Metod funktsii Lyapunova i ego prilozheniya” (Krym, Alushta, 5–12 sentyabrya 1998 g.), Izd-vo Simferopolskogo gos. un-ta, Simferopol, 1998, 80

[417] Shamolin M. V., “Many-dimensional topographical Poincare systems in rigid body dynamics”, Abstracts of GAMM Wissenschaftliche Jahrestagung' 98 (6–9 April 1998), Universität Bremen, Bremen, Germany, 1998, 128

[418] Shamolin M. V., “New two-parametric families of the phase portraits in three-dimensional rigid body dynamics”, Mezhdunar. konf., posvyaschënnaya 90-letiyu so dnya rozhdeniya L. S. Pontryagina, Tez. dokl. Differents. uravn. (Moskva, 31 avgusta – 6 sentyabrya 1998 g.), Izd-vo Mosk. un-ta, M., 1998, 97–99

[419] Shamolin M. V., “Some classical problems in a three-dimensional dynamics of a rigid body interacting with a medium”, Proc. of ICTACEM' 98 (Kharagpur, India, December 1–5, 1998), Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India, 1998, 11

[420] Shamolin M. V., “Integrability in terms of transcendental functions in rigid body dynamics”, Book of Abstracts of GAMM Annual Meeting (Metz, France, April 12–16, 1999), Université de Metz, Metz, France, 1999, 144

[421] Shamolin M. V., “Long-periodic trajectories in rigid body dynamics”, Sixth Colloquium on the Qualitative Theory of Differential Equations (Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, August 10–14, 1999), Szeged, Hungary, 1999, 47

[422] Shamolin M. V., “Mathematical modelling in 3D dynamics of a rigid interacting with a medium”, Book of Abstracts of the Second Int. Conf. “Tools for Mathematical Modelling” (Saint-Petersburg, Russia, 14–19 June 1999), Saint-Petersburg State Tech. Univ., Saint-Petersburg, 1999, 122–123

[423] Shamolin M. V., “Methods of analysis of a deceleration of a rigid in 3D medium”, Contributed Abstracts of 3rd ENOC (Copenghagen (Lyngby), Denmark, August 8–12, 1999), Tech. Univ. of Denmark, Copenghagen, 1999

[424] Shamolin M. V., “New families of the non-equivalent phase portraits in 3D rigid body dynamics”, Abstracts of Second Congress ISAAC 1999 (Fukuoka, Japan, August 16–21, 1999), Fukuoka Ins. of Tech., Fukuoka, 1999, 205–206

[425] Shamolin M. V., “Properties of integrability of systems in terms of transcendental functions”, Final Progr. and Abstracts of Fifth SIAM Conf. on Appl. of Dynam. Syst. (Snowbird, Utah, USA, May 23–27, 1999), SIAM, 1999, 60

[426] Shamolin M. V., “Some properties of transcendental integrable dynamical systems”, Book of Abstracts of EQUADIFF 10 (Berlin, August 1–7, 1999), Free Univ. of Berlin, Berlin, 1999, 286–287

[427] Shamolin M. V., “Structural stability in 3D dynamics of a rigid body”, WCSMO–3, Short Paper Proc., Vol. 2 (Buffalo, NY, May 17–21, 1999), State Univ. of NY at Buffalo, Buffalo, 1999, 475–477

[428] 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

[429] Shamolin M. V., “About interaction of a rigid body with a resisting medium under an assumption of a jet flow”, Book of Abstracts II (General sessions) of 4th EUROMECH Solid Mech. Conf. (Metz, France, June 26–30, 2000), Univ. of Metz, Metz, 2000, 703

[430] Shamolin M. V., “Integrability and non-integrability in terms of transcendental functions”, CD-Abstracts of 3rd ECM, Poster sessions (Barcelona, Spain, June 10–14, 2000), Poster No 36

[431] Shamolin M. V., “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability”, Book of Abstracts of ECCOMAS 2000 (Barcelona, Spain, 11–14 September 2000), Barcelona, 2000, 495

[432] 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, Spain, 11–14 September 2000), Barcelona, 2000 | Zbl

[433] Shamolin M. V., “Methods of analysis of dynamics of a rigid body interacting with a medium”, Book of Abstracts of Annual Sci. Conf. GAMM 2000 (Univ. of Göttingen, 2–7 April, 2000), Univ. of Göttingen, Göttingen, 2000, 144

[434] Shamolin M. V., “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium”, Book of Abstracts of 16th IMACS World Congress 2000 (Lausanne, Switzerland, August 21–25, 2000), EPFL, Lausanne, Switzerland, 2000, 283

[435] Shamolin M. V., “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium”, CD-Proc. of 16th IMACS World Congress 2000 (Lausanne, Switzerland, August 21–25, 2000), EPFL, Lausanne, Switzerland, 2000

[436] Shamolin M. V., “Comparison of some cases of integrability in dynamics of a rigid body interacting with a medium”, Book of Abstracts of Annual Sci. Conf. GAMM 2001 (ETH, Zurich, 12–15 February 2001), ETH, Zurich, 2001, 132

[437] Shamolin M. V., “Pattern recognition in the model of the interaction of a rigid body with a resisting medium”, Col. of Abstracts of First SIAM–EMS Conf. “Applied Mathematics in our Changing World” (Berlin, Germany, September 2–6, 2001), Springer, Birkhäuser, 2001, 66

[438] Shamolin M. V., “Dynamical systems with the variable dissipation in 3D dynamics of a rigid body interacting with a medium”, Book of Abstracts of 4th ENOC (Moscow, Russia, August 19–23, 2002), Inst. Probl. Mech. Russ. Acad. Sci., Moscow, 2002, 109

[439] Shamolin M. V., “Foundations in diferential and topological diagnostics”, Book of Abstracts of Annual Sci. Conf. GAMM 2002 (Univ. of Augsburg, March 25–28, 2002), Univ. of Augsburg, Augsburg, 2002, 154

[440] Shamolin M. V., “Methods of analysis of dynamics of a 2D- 3D-, or 4D-rigid body with a medium”, Abstracts, Short Communications, Poster Sessions of ICM–2002 (Beijing, August 20–28, 2002), Higher Education Press, Beijing, China, 2002, 268

[441] 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

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

[443] Shamolin M. V., “Global structural stability in dynamics of a rigid body interacting with a medium”, 5th ICIAM (Sydney, Australia, 7–11 July 2003), Univ. of Technology, Sydney, 2003, 306

[444] Shamolin M. V., “Integrability and nonintegrability in terms of transcendental functions”, Book of Abstracts of Annual Sci. Conf. GAMM 2003 (Abano Terme–Padua, Italy, 24–28 March 2003), Univ. of Padua, Padua, 2003, 77

[445] 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

[446] Shamolin M. V., “Some questions of differential and topological diagnostics”, Book of Abstracts of 5th European Solid Mech. Conf., ESMC–5 (Thessaloniki, Greece, August 17–22, 2003), Aristotle Univ. Thes. (AUT), European Mech. Sci. (EUROMECH), 2003, 301

[447] 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

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

[449] Shamolin M. V., “Mathematical model of interaction of a rigid body with a resisting medium in a jet flow”, Abstracts. 76 Annual Sci. Conf., GAMM, Part 1. (Luxembourg, March 28 – April 1, 2005), Univ. du Luxembourg, Luxembourg, 2005, 94–95

[450] 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), Aberdeen, 2005, 112

[451] Shamolin M. V., “Structural stable vector fields in rigid body dynamics”, Abstracts of 8th Conf. on Dynamical Systems (Theory and Applications), DSTA 2005 (Lodz, Poland, December 12–15, 2005), Tech. Univ. Lodz, Lodz, 2005, 78

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

[453] Shamolin M. V., “Almost conservative systems in dynamics of a rigid body”, Book of Abstracts, 77th Annual Meeting of GAMM (Technische Univ. Berlin, March 27–31, 2006), Technische Univ. Berlin, Berlin, 2006, 74

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

[455] Shamolin M. V., “On the problem of a symmetric body motion in a resisting medium”, Book of Abstracts of EMAC–2007 (Hobart, Australia, 1–4 July 2007), Univ. Tasmania, Hobart, Australia, 2007, 25

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

[457] Shamolin M. V., “The cases of integrability in 2D-, 3D-, and 4D-rigid body”, Abstracts of Short Communications and Posters of Int. Conf. “Dynamical Methods and Mathematical Modelling” (Valladolid, Spain, September 18–22, 2007), ETSII, Valladolid, 2007, 31

[458] Shamolin M. V., “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium”, Abstracts of 9th Conf. on Dynamical Systems (Theory and Applications), DSTA 2007 (Lodz, Poland, December 17–20, 2007), Tech. Univ. Lodz, Lodz, 2007, 115

[459] 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, Vol. 1 (Lodz, Poland, December 17–20, 2007), Tech. Univ. Lodz, Lodz, 2007, 415–422

[460] Shamolin M. V., Shebarshov D. V., “Lagrange tori and the Hamilton–Jacobi equation”, Book of Abstracts of Conf. “Partial Differential Equations: Theory and Numerical Solutions” (Prague, August 10–16, 1998), Charles University, Prague, Czech Rep., 1998, 88

[461] Weyher C. L., “Observations sur le vol plane par obres”, L'Aeronaute, 1890