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Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconservative Force Fields

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Abstract

This paper is a survey of integrable cases in dynamics of two-, three-, and four-dimensional rigid bodies 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.

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  175. N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a one-phase problem with free boundary,” in: Materials of Voronezh Winter Mathematical School ‘Contemporary Methods of Function Theory and Related Problems,’ Voronezh, January 26–February 1, 2011 [in Russian], Voronezh State University, Voronezh (2011), p. 307.

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  189. M. V. Shamolin, “A new two-parameter family of phase portraits with limit cycles in the dynamics of a rigid body interacting with a medium,” in: Modelling and Study of Stability of Systems, Scientific Conference, May 15–19, 1995. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1995), p. 125.

  190. M. V. Shamolin, “On relative stability of dynamical systems in the problem of the motion of a body in a resisting medium,” in: Abstracts of Reports of Chebyshev Readings, Vestn. VGU, Ser. 1, Mat., Mekh., 6, 17 (1995).

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  194. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” in: II Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 25–30, 1996. Abstracts of Reports. Pt. III [in Russian], Novosibirsk (1996), p. 267.

  195. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197.

  196. M. V. Shamolin, “On a certain integrable case in dynamics of spatial body motion in a resisting medium,” in: II Symposium in Classical and Celestial Mechanics. Abstracts of Reports. Velikie Luki, August 23–28, 1996 [in Russian], Moscow–Velikie Luki (1996), pp. 91–92.

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  200. M. V. Shamolin, “A list of integrals of dynamical equations in the spatial problem of the motion of a body in a resisting medium,” in: Modelling and Study of Stability of Systems, Scientific Conference, May 20–24, 1996. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1996), p. 142.

  201. M. V. Shamolin, “Jacobi integrability of problem of a spatial pendulum placed in a flow of a medium,” in: Modelling and Study of Systems, Scientific Conference, May, 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 143.

  202. M. V. Shamolin, “Qualitative methods in dynamics of a rigid body interacting with a medium,” in: YSTM96: ‘Young Peoples, the Third Millenium,’ Proceedings of International Congress (Ser. Professional) [in Russian], 2, NTA “APFN,” Moscow (1997), pp. I-4.

  203. M. V. Shamolin, “Mathematical modelling of dynamics of a spatial pendulum in a flow of a medium,” in: Proceedings of VII International Symposium ‘Methods of Discrete Singularities in Problems of Mathematical Physics,’ June 26–29, Feodociya [in Russian], Kherson State Technical University, Kherson (1997), pp. 153–154.

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  208. M. V. Shamolin, “Absolute and relative structural stability in spatial dynamics of a rigid body interacting with a medium,” in: Proceedings of International Conference ‘Mathematics in Inductry’, ICIM–98, Taganrog, June 29– July 03, 1998 [in Russian], Taganrog State Pedagogical Institute, Taganrog (1998), pp. 332–333.

  209. M. V. Shamolin, “Qualitative and numerical methods in some problems of spatial dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of 5th International Conference-Workshop ‘Engineering-Physical Problems of New Techniques,’ Moscow, May 19–22, 1998 [in Russian], Moscow State Technical University, Moscow (1998), pp. 154–155.

  210. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of International Conference ‘Nonlinear Analysis and Its Applications’ (Moscow, September 1–5, 1998) [in Russian], Moscow (1998), p. 131.

  211. M. V. Shamolin, “Some problems of spatial dynamics of a rigid body interacting with a medium under quasi-stationarity conditions,” in: Abstracts of Reports of All-Russian Scientific-Technical Conference of Young Scientists ‘Modern Problems of Aero-Cosmic Science,’ Zhukovskii, May 27–29, 1998 [in Russian], Central Aero-Hydrodynamical Institute, Moscow (1998), pp. 89–90.

  212. M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: CD-Proceedings of the Congress ‘Nonlinear Analysis and Its Applications’, Moscow, Russia, Sept. 1–5, 1998 [in Russian], Moscow (1999), pp. 497–508.

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  215. M. V. Shamolin, “Families of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 6, 29–37 (1998).

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  220. M. V. Shamolin, “On roughness of dissipative systems and relative roughness and non-roughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999).

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  221. M. V. Shamolin, “Problem of the motion of a four-dimensional body in a resisting medium and one case of integrability,” in: Book of Abstracts of the Third International Conference ‘Differential Equations and Applications’, St.-Petersburg, Russia, June 12–17, 2000 [in Russian], St.-Petersburg State University, St.-Petersburg (2000), p. 198.

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  224. M. V. Shamolin, “Many-dimensional Poincaré systems and transcendental integrability,” in: IV Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 26–July 01, 2000. Abstracts of Reports, Pt. I. [in Russian], Novosibirsk, Institute of Mathematics (2000), pp. 25–26.

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  226. M. V. Shamolin, “On a certain case of Jacobi integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of International Conference in Differential and Integral Equations, Odessa, September 12–14, 2000 [in Russian], AstroPrint, Odessa (2000), pp. 294–295.

  227. M. V. Shamolin, “On roughness of dissipative systems and relative roughness of variable dissipation systems,” in: Abstracts of Reports of P. K. Rashevskii Workshop in Vector and Tensor Analysis, Vestn. MGU, Ser. 1, Mat., Mekh., 2, 63 (2000).

  228. M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000).

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  229. M. V. Shamolin, “Comparison of certain integrability cases from two-, three-, and fourdimensional dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of V Crimean International Mathematical School ‘Lyapunov function Method and Its Application,’ (MLF–2000), Crimea, Alushta, September 5–13, 2000 [in Russian], Simpheropol’ (2000), p. 169.

  230. M. V. Shamolin, “Integrability of a problem of four-dimensional rigid body in a resisting medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Fund. Prikl. Mat., 7, No. 1, 309 (2001).

  231. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” in Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Fund. Prikl. Mat., 7, No. 1, 302–303 (2001).

  232. M. V. Shamolin, “New Jacobi integrable cases in dynamics of two-, three-, and four-dimensional rigid body interacting with a medium,” Absracts of Reports of VIII All-Russian Congress in Theoretical and Applied Mechanics, Perm’, August 23–29, 2001 [in Russian], Ural Department of Russian Academy of Sciences, Ekaterinburg (2001), pp. 599–600.

  233. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Scientific Conference, May 22–25, 2001 [in Russian], Kiev (2001), p. 344.

  234. M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela 1, 189–193 (2001).

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  238. M. V. Shamolin, “On integrability of certain classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002).

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  239. M. V. Shamolin, “Integrability in transcendental functions in rigid body dynamics,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April 17–27, 2003, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2003), p. 130.

  240. M. V. Shamolin, “On integrability of nonconservative dynamical systems in transcendental functions,” in: Modelling and Study of Stability of Systems, Scientific Conference, May 27–30, 2003, Abstracts of Reports [in Russian], Kiev (2003), p. 377.

  241. M. V. Shamolin, “On a certain spatial problem of rigid body motion in a resisting medium,” in: Abstracts of Reports of International Scientific Conference Third Polyakhov Readings,’ St.-Petersburg, February 4–6, 2003 [in Russian], NIIKh St.-Petersburg Univ, (2003), pp. 170–171.

  242. M. V. Shamolin, “Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004).

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  244. M. V. Shamolin, “Integrability in transcendental functions in rigid body dynamics,” in: Mathematical Conference ‘Modern Problems of Applied Mathematics and Mathematical Modelling,’ Voronezh, December 12–17, 2005 [in Russian], Voronezh State Academy, Voronezh (2005), p. 240.

  245. M. V. Shamolin, “Integrability of nonconservative systems in elementary functions,” in: X Academician M. Kravchuk Mathematical International Conference, September 3–15, 2004, Kiev [in Russian], Kiev (2004), p. 279.

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  247. M. V. Shamolin, “On a certain integrable case in dynamics on so(4) × ℝ4,” in: Abstracts of Reports of All-Russian Conference ‘Differential Equations and Their Applications,’ (SamDif–2005), Samara, June 27–July 2, 2005 [in Russian], Univers-Grupp, Samara (2005), pp. 97–98.

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  249. M. V. Shamolin, “On the motion of a rigid body in a resisting medium with account for rotational derivatives of aerodynamical force moment in angular velocity,” in: Modelling and Studying of Systems, Scientific Conference, May 23–25, 2005. Abstracts of Reports [in Russian], Kiev (2005), p. 351.

  250. M. V. Shamolin, “On the motion of a body in a resisting medium with account for rotational derivatives of aerodynamical force moment in angular velocity,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings–2005,’ Sec. Mechanics, April, 2005, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2005), p. 182.

  251. M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of International Conference ‘Functional Spaces, Approximation Theory, and Nonlinear Analysis’ Devoted to the 100th Anniversary of A. M. Nikol’skii, Moscow, May 23–29, 2005 [in Russian], V. A. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow (2005), p. 244.

  252. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium with account for rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4. 482–485 (2005).

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  253. M. V. Shamolin, “Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005).

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  254. M. V. Shamolin, “Problem on rigid body spatial drag in a resisting medium,” Izv. Ross. Akad. Nauk, Mekhanika Tverdogo Tela, 3, 45–57 (2006).

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  255. M. V. Shamolin, “On the spatial problem of rigid body interaction with a resisting medium,” in: Abstracts of Reports of IX All-Russian Congress in Theoretical and Applied Mechanics, Nizhnii Novgorod, August 22–28, 2006. Vol. I [in Russian]. N. I. Lobachevskii Nizhegorodskii State Univesity, Niznii Novgorod (2006), p. 120.

  256. M. V. Shamolin, “Model problem of the motion of a body in a resisting medium with account for dependence of resistance force on angular velocity,” in: Scientifuc Report of Institute of Mechanics, Moscow State University [in Russian], No. 4818, Institute of Mechanics, Moscow State University, Moscow (2006).

  257. M. V. Shamolin, “On a case of complete integrability in four-dimensional rigid body dynamics,” Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Vladimir, July 10–15, 2006 [in Russian], Vladimir State University, Vladimir (2006), pp. 226–228.

  258. M. V. Shamolin, “On trajectories of characteristic points of a rigid body moving in a medium,” in: International Conference ‘Fifth Okunev Readings,’ St.-Petersburg, June 26–30, 2006. Abstracts of Reports [in Russian], Balt. State Technical University, St. Petersburg (2006), p. 34.

  259. M. V. Shamolin, “Spatial problem on the motion of a rigid body in a resisting medium,” in: VIII Crimean International Mathematical School ‘Lyapunov Function Method and Its Applications,’ Abstracts of Reports, Alushta, September 10–17, 2006, Tavriya National University [in Russian], DiAiPi, Simpheropol’ (2006), p. 184.

  260. M. V. Shamolin, “Variable dissipation systems in dynamics of the interacting of a rigid body with a medium,” Fourth Polyakhov Readings, Abstracts of Reports of International Scientific Conference in Mechanics, St.-Petersburg, February 7–10, 2006 [in Russian], VVM, St.-Petersburg (2006), p. 86.

  261. M. V. Shamolin, “On account of rotational derivatives of an aerodynamical force moment on the motion of a body in a resisting medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 44.

  262. M. V. Shamolin, “Integrability in elementary functions of variable dissipation systems,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 38.

  263. M. V. Shamolin, “Integrability of problem of the motion of a four-dimensional rigid body in a resisting medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 21.

  264. M. V. Shamolin, “Integrability of strongly nonconservative systems in transcendental elementary functions,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 40.

  265. M. V. Shamolin, Methods for Analysis of Variable-Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007).

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  266. M. V. Shamolin, “Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 17.

  267. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], 2nd Corrected and Enlarged Edition, Ekzamen, Moscow (2007).

  268. M. V. Shamolin, “Some model problems of dynamics for a rigid body interacting with a medium,” Prikl. Mekh., 43, No. 10, 49–67 (2007).

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  269. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 27.

  270. M. V. Shamolin, “On integrability in transcendental functions,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23, (2007), p. 34.

  271. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Modelling and Study of Systems, Scientific Conference, May 22–25, 2007. Abstracts of Reports [in Russian], Kiev (2007), p. 249.

  272. M. V. Shamolin, “On integrability of motion of four-dimensional body-pendulum situated in a flow of a medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 37.

  273. M. V. Shamolin, “On stability of a certain regime of rigid body motion in a resisting medium,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings-2007,’ Sec. Mechanics, Moscow, Moscow State University, April, 2007 [in Russian], MGU, Moscow (2007), p. 153.

  274. M. V. Shamolin, “On account of rotational derivatives of aerodynamical force moment on body motion in a resisting medium,” in: Abstracts of Session of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Problems in Mathematics, Fundamental Directions [in Russian], 23, Moscow (2007), p. 26.

  275. M. V. Shamolin, “On rigid body motion in a resisting medium taking account of rotational derivatives of aerodynamical force moment in angular velocity,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions [in Russian], 23 (2007), p. 44.

  276. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in a flowing medium taking account of rotational derivatives of moments of its action force,” Izv. Ross Akad. Nauk, Mekhanika Tverdogo Tela, 3, 187–192 (2007).

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  277. M. V. Shamolin, “Cases of complete integrability in dynamics of a rigid body interacting with a medium,” Abstracts of Reports of All-Russiann Conference ‘Modern Problems of Contionuous Medium Mechanics’ Devoted to the Memory of L. I. Sedov in Connection With His 100th Anniversary, Moscow, November, 12–14, 2007 [in Russian], MIAN, Moscow (2007), pp. 166–167.

  278. M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body in a nonconservative force field,” in: Abstract of Reports of International Conference ‘Analysis and Singularities,’ Devoted to 70th Anniversary of V. I. Arnol’d, August 20–24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110–112.

  279. M. V. Shamolin, “Cases of complete integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Abstracts of Reports of International Conference ‘Classical Problems of Rigid Body Dynamics,’ June 9–13, 2007 [in Russian], Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (2007), pp. 81–82.

  280. M. V. Shamolin, “Case of complete integrability in dynamics of a four-dimensional rigid body in nonconservative force field,” in: ‘Nonlinear Dynamical Analysis-2007,’ Abstracts of Reports of International Congress, St. Petersburg, June 4–8, 2007 [in Russian], St.-Petersburg State University, St.-Petersburg (2007), p. 178.

  281. M. V. Shamolin, “A case of complete integrability in dynamics on a tangent bundle of twodimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007).

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  282. M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications,” Fund. Prikl. Mat., 14, No. 3, 3–237 (2008).

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  283. M. V. Shamolin, “Qualitative methods of analysis of variable dissipation systems in Dynamics,” in: International Conference ‘Sixth Okunev Readings,’ St.-Petersburg, June 23–27, 2008. Materials of Reports, Vol. III [in Russian], Balt. State Technical University, St.-Petersburg (2008), pp. 34–39.

  284. M. V. Shamolin, “Methods of analysis of dynamical systems with sign-variable dissipation,” in: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, June 26–July 2, 2008 [in Russian], Vladimir, Vladimir State University (2008), pp. 259–260.

  285. M. V. Shamolin, “Methods of analysis of dynamical systems with certain group of symmetry,” in: Abstracts of Reports of International Conference ‘Differential Geometry and Topology’ in Honor of 100th Birthday of L. S. Pontryagin, Moscow, June 17–22, 2008, Faculty of VMK at MSU, Paks-Press, Moscow (2008), pp. 208–209.

  286. M. V. Shamolin, “Comparison of certain integrability cases from two-, three-, and fourdimensional dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of IX Crimean International Mathematical School ‘Lyapunov Function Method and Its Application,’ Crimea, Alushta, September 15–20, 2008 [in Russian], Simpheropol’ (2008), pp. 181–182.

  287. M. V. Shamolin, “New integrable cases in dynamics of a body interacting with a medium with allowance for dependence of resistance force moment on angular velocity,” Prikl. Mat. Mekh., 72, No. 2, 273–287 (2008).

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  288. M. V. Shamolin, “New cases of complete integrability in dynamics of symmetric four-dimensional rigid body in nonconservative field,” in: Materials of International Conference ‘Contemporary Problems of Mathematics, Mechanics, and Informatics’ in Honor of 85th Birthday of L. A. Tolokonnikov, Tula, Russia, November 17–21, 2008 [in Russian], Grif and Ko., Moscow (2008), pp. 317–320.

  289. M. V. Shamolin, “New integrable case in dynamics of four-dimensional rigid body in nonconservative field of forces,” in: Materials of Voronezh Spring Mathematical School ‘Pontryagin Readings–XIX,’ Voronezh, May, 2008 [in Russian], Voronezh State University, Voronezh (2008), pp. 231–232.

  290. M. V. Shamolin, “Integrability of some classes of dynamical systems in terms of elementary functions,” Vestn. MGU, Ser. 1, Mat., Mekh., 3, 43–49 (2008).

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  291. M. V. Shamolin, “Systems with sign-variable dissipation in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April, 2008, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2008), pp. 159–160.

  292. M. V. Shamolin, “Three-parameter family of phase portraits in dynamics of a solid interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008).

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  293. M. V. Shamolin, “Diagnosis of failures in certain non-direct control system,” Elektronnoe Modelirovanie, 31, No. 4, 55–66 (2009).

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  294. M. V. Shamolin, “Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field,” in: Contemporary Mathematics and Its Applications [in Russian], 65, Mathematical Physics, Combinatorics, and Optimal Control (2009), pp. 132–142.

  295. M. V. Shamolin, “Methods for analysis of various dissipation dynamical systems,” in: Abstract of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary athematics and Its Applications [in Russian], 62, Geometry and Mechanics (2009), p. 13.

  296. M. V. Shamolin, “New cases of integrability in dynamics of four-dimensional rigid body in a nonconservative field,” in: Abstract of sessions of workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 65, Mathematical Physics, Combinatorics, and Optimal Control (2009), p. 6.

  297. M. V. Shamolin, “Certain cases of complete integrability in spatial dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of International Scientific Conference Fifth Polyakhov Readings,’ St.-Petersburg, February 3–6, 2009 [in Russian], St.-Petersburg Univ (2009), p. 73.

  298. M. V. Shamolin, “Certain cases of complete integrability in spatial dynamics of a rigid body interacting with a medium,” in: Proc. of International Scientific Conference Fifth Polyakhov Readings,’ St.-Petersburg, February 3–6, 2009 [in Russian], St.-Petersburg Univ, (2009), pp. 144–150.

  299. M. V. Shamolin, “New cases of full integrability in dynamics of a dynamically symmetric fourdimensional solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 425, No. 3, 338–342 (2009).

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  300. M. V. Shamolin, “New cases of complete integrability in spatial dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April, 2009, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2009), pp. 153–154.

  301. M. V. Shamolin, “On integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Materials of Voronezh Spring Mathematical School ‘Pontryagin Readings–XX,’ Voronezh, May 3–9, 2009 [in Russian], Voronezh State University, Voronezh (2009), pp. 191–192.

  302. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Contemporary Mathematics and Its Applications. [in Russian], 62, Geometry and Mechanics (2009), pp. 131–171.

  303. M. V. Shamolin, “On integrability of certain classes of dynamical systems,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 62, Geometry and Mechanics (2009), p. 10.

  304. M. V. Shamolin, “On stability of certain conditions of rigid body motion in a resisting medium,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 62, Geometry and Mechanics (2009), pp. 10–11.

  305. M. V. Shamolin, “Stability of rectilinear translational motion,” Prikl. Mekh., 45, No. 6, 125–140 (2009).

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  306. M. V. Shamolin, “On trajectories diverging to infinity for planar dynamical systems,” in: Abstract of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 62, Geometry and Mechanics (2009), p. 7.

  307. M. V. Shamolin, “Generalized problem of differential diagnosis and its possible solution,” Elektronnoe Modelirovanie, 31, No. 1, 97–115 (2009).

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  308. M. V. Shamolin, “Solution of diagnosis problem in case of precise trajectory measurements with error,” Elektronnoe Modelirovanie, 31, No. 3, 73–90 (2009).

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  309. M. V. Shamolin, “Variable dissipation systems: methods, approaches, and applications,” in: Abstracts of Reports of Scientific Conference, May 27–29, 2009 [in Russian], Kiev (2009), p. 163.

  310. M. V. Shamolin, “Cases of integrability of equations of motion of a four-dimensional rigid body in a nonconservative field of forces,” in: Materials of International Conference ‘Contemporary Problems in Mathematics, Mechanics, and Its Applications’ Devoted to the 70th Anniversary of V. A. Sadovnichii, Moscow, March 30–April 2, 2009 [in Russian], Universitetskaya Kniga, Moscow (2009), p. 233.

  311. M. V. Shamolin, “Case of complete integrability in Dynamics of symmetric four-dimensional rigid body in a nonconservative field,” in: Abstract of sessions of workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 65, Mathematical Physics, Combinatorics, and Optimal Control (2009), p. 9.

  312. M. V. Shamolin, “Motion diagnosis of aircraft in mode of planned descent,” Elektronnoe Modelirovanie, 32, No. 5, 31–44 (2010).

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  313. M. V. Shamolin, “Diagnosis of a system of direct control of aircraft motion,” Elektronnoe Modelirovanie, 32, No. 1, 45–52 (2010).

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  314. M. V. Shamolin, “Integrability and non-integrability of dynamical systems in transcendental functions,” in: Abstracts of Reports of Voronezh Winter Mathematical School of S. G. Kreyn, Voronez, 2010 [in Russian], Voronezh State University, Voronezh (2010), pp. 159–160.

  315. M. V. Shamolin, “On the problem of the motion of the body with front flat butt end in a resisting medium,” in: Scientific Report of Institute of Mechamics, Moscow State University [in Russian], No. 5052, Institute of Mechanics, Moscow State University, Moscow (2010).

  316. M. V. Shamolin, “New cases of integrability in the spatial dynamics of a rigid body,” Dokl. Ross. Akad. Nauk, 431, No. 3, 339–343 (2010).

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  317. M. V. Shamolin, “Spatial motion of a rigid body in a resisting medium,” Prikl. Mekh., 46, No. 7, 120–133 (2010).

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  318. M. V. Shamolin, “Cases of complete integrability of the equations of motion of a dynamicalsymmetric four-dimensional rigid body in a nonconservative field,” in: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, July 2–7, 2010 [in Russian], Vladimir, Vladimir State University (2010), p. 195.

  319. M. V. Shamolin, “Cases of complete integrability of the spatial motion equations of a rigid body in a resisting medium,” in: Abstracts of Reports of XI International Conference ‘ Stability and Oscillations of Nonlinear Control Systems,’ Moscow, IPU RAN, June 1–4, 2010 [in Russian], Moscow, IPU RAN (2010), pp. 429–431.

  320. M. V. Shamolin, “Cases of complete integrability of spatial dynamics equations of a rigid body in a resisting medium,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April, 2010, Moscow , M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2010), p. 172.

  321. M. V. Shamolin, “A completely integrable case in the dynamics of a four-dimensional rigid body in a non-conservative field,” Usp. Mat. Nauk, 65, No. 1, 189–190 (2010).

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  322. M. V. Shamolin, “Rigid body motion in a resisting medium,” Matem. Mod., 23, No. 12, 79–104 (2011).

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  323. M. V. Shamolin, “Diagnosis of gyro-stabilized platform included in control system of aircraft motion,” Elektronnoe Modelirovanie, 33, No. 3, 121–126 (2011).

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  324. M. V. Shamolin, “Dynamical invariants of integrable variable dissipation dynamical systems ,” Vestnik Nizhegorod. Univ., 2, No. 4, 356–357 (2011).

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  325. M. V. Shamolin, “A multiparameter family of phase portraits in the dynamics of a rigid body interacting with a medium,” Vestn. MGU, Ser. 1, Mat., Mekh., 3, 24–30 (2011).

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  326. M. V. Shamolin, “A new case of integrability in dynamics of a 4D-solid in a nonconservative field,” Dokl. Ross. Akad. Nauk, 437, No. 2, 190–193 (2011).

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  327. M. V. Shamolin, “New case of complete integrability of the dynamical equations on the tangential stratification of three-dimensional sphere,” in: Vestnik SamGU. Natural Sciences, No. 5(86), 187–189, (2011).

  328. M. V. Shamolin, “Complete lists of first integrals in dynamics of four-dimensional rigid body in a nonconservative force,” in: Abstracts of Reports of International Conference Devoted to 110th Anniversary of I. G. Petrovskii, 2011, Moscow [in Russian], MGU and ‘Intuit. RU’, Moscow (2011), pp. 389–390.

  329. M. V. Shamolin, “Complete list of first integrals in the problem on the motion of a 4D solid in a resisting medium under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 440, No. 2, 187–190 (2011).

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  330. M. V. Shamolin, “Comparison of complete integrability cases from two-, three-, and fourdimensional dynamics of a rigid body in a nonconservative field,” in: Abstracts of Reports of Scientific Conference ‘Dynamical System Modelling and Stability Investigation’, May 25–27, 2011 [in Russian], Kiev (2011), p. 139.

  331. M. V. Shamolin, “The problem of a rigid body motion in a resisting medium with the assumption of dependence of the force moment on the angular velocity,” Matem. Mod., 24, No. 10, 109–132 (2012).

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  332. M. V. Shamolin, “Variety of cases of integrability in rigid body dynamics in a nonconservative field,” in: Abstracts of Reports of Scientific Conference ‘Lomonosov Readings,’ Sec. Mechanics, April, 2012, Moscow, M. V. Lomonosov Moscow State University [in Russian], MGU, Moscow (2012), p. 156.

  333. M. V. Shamolin, “Some questions of qualitative theory in dynamics of systems with the variable dissipation,” in: Contemporary Mathematics and Its Applications [in Russian], 78, Partial Differential Equations and Optimal Control (2012), pp. 138–147.

  334. M. V. Shamolin, “New cases of integrability in transcendental functions in rigid body dynamics in a nonconservative field,” in: Materials of Voronezh Spring Mathematical School ‘Pontryagin Readings–XXIII,’ Voronezh, May 3–9, 2012 [in Russian], Voronezh State University, Voronezh (2012), p. 200.

  335. M. V. Shamolin, “A new case of integrability in the dynamics of a 4D-rigid body in a nonconservative field under the assumption of linear damping,” Dokl. Ross. Akad. Nauk, 444, No. 5, 506–509 (2012).

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  336. M. V. Shamolin, “A new case of integrability in spatial dynamics of a rigid solid interacting with a medium under assumption of linear damping,” Dokl. Ross. Akad. Nauk, 442, No. 4, 479–481 (2012).

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  337. M. V. Shamolin, “New case of integrability in transcendental functions in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of XII International Conference ‘Stability and Oscillations of Nonlinear Control Systems,’ Moscow, IPU RAN, June 5–8, 2012 [in Russian], Moscow, IPU RAN (2012), pp. 339–341.

  338. M. V. Shamolin, “Review of cases of integrability in dynamics of small- and multi-dimensional rigid body in a nonconservative field,” in: Abstracts of Reports of International Conference in Differential Equations and Dynamical Systems, Suzdal’, June 26–July 4, 2012 [in Russian], Suzdal’, Kollektiv Avtorov (2012), pp. 179–180.

  339. M. V. Shamolin, “Complete list of first integrals of dynamical equations of the spatial motion of a rigid body in a resisting medium under assumption of linear damping,” Vestn. MGU, Ser. 1, Mat., Mekh., 4, 44–47 (2012).

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  340. M. V. Shamolin, “Systems with variable dissipation: Methods, approaches, and applications,” in: Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 76, Geometry and Mechanics (2012), p. 6.

  341. M. V. Shamolin, “Cases of integrability in dynamics of four-dimensional rigid body in a nonconservative field,” in: Materials of Voronezh Winter Mathematical School of S. G. Kreyn, Voronez, January 25–30, 2012 [in Russian], Voronezh State University, Voronezh (2012), pp. 213–215.

  342. M. V. Shamolin, “Cases of integrability in spatial dynamics of a rigid body in a medium in a jet flow,” in: Abstracts of Reports of International Scientific Conference ‘Sixth Polyakhov Readings,’ St.-Petersburg, January 31–February 3, 2012 [in Russian], I. V. Balabanov Publisher, St.-Petersburg (2012), p. 75.

  343. M. V. Shamolin, “Cases of integrability in spatial dynamics of a rigid body interacting with a medium under assumption of linear damping,” in: Proc. of X International Chetaev Conference ‘Analytical Mechanics, Stability and Control,’ Kazan’, Russia, June 12–16, 2012 [in Russian], Kazan’ State Technical University, Kazan’ (2012), pp. 508–514.

  344. M. V. Shamolin, “Cases of complete integrability in transcendental functions in Dynamics of a rigid body interacting with a medium,” in: Abstract of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics’, Contemporary Mathematics and Its Applications [in Russian], 76, Geometry and Mechanics (2012), p. 7.

  345. M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,” in: Contemporary Mathematics and Its Applications [in Russian], 76, Geometry and Mechanics (2012), pp. 84–99.

  346. M. V. Shamolin and S. V. Tsyptsyn, “Analytical and numerical study of trajectories of the motion of a body in a resisting medium,” in: Scientific Report of Institute of Mechanivs, Moscow State University [in Russian], No. 4289, Institute ofMechanics, Moscow State University, Moscow (1993).

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

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

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

  350. M. V. Shamolin, “Qualitative methods to the dynamical model of an interaction of a rigid body with a resisting medium and new two-parameter families of the phase portraits,” in: DynDays’95 (Sixteenth Annual Informal Workshop), Program and Abstracts, Lyon, June 28–July 1, 1995, Lyon, France (1995), p. 185.

  351. M. V. Shamolin, “Poisson-stable and dense orbits in rigid body dynamics,” in: 3rd Experimental Chaos Conference, Advance Program, Edinburg, Scotland, August 21–23, 1995, Edinburg, Scotland (1995), p. 114.

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

  353. M. V. Shamolin, “Qualitative methods in interacting with the medium rigid body dynamics,” in: Abstracts of GAMM Wissenschaftliche Jahrestangung’96, 27–31 May, 1996, Prague, Czech Rep., Karls-Universit¨at Prague (1996), pp. 129–130.

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

  355. M. V. Shamolin, “Relative structural stability and relative structural instability of different degrees in topological dynamics,” in: Abstracts of International Topological Conference Dedicated to P. S. Alexandroff’s 100th Birthday ‘Topology and Applications’, Moscow, May 27–31, 1996, Phasys, Moscow (1996), pp. 207–208.

  356. M. V. Shamolin, “Topographical Poincaré systems in many dimensional spaces,” in: 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), p. 45.

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

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

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

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

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 125, Dynamical Systems, 2013.

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Shamolin, M.V. Variety of Integrable Cases in Dynamics of Low- and Multi-Dimensional Rigid Bodies in Nonconservative Force Fields. J Math Sci 204, 379–530 (2015). https://doi.org/10.1007/s10958-014-2209-0

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