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Integrable Variable Dissipation Systems on the Tangent Bundle of a Multi-Dimensional Sphere and Some Applications

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Abstract

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 energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the 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 to 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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  122. 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 (1996).

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  123. M. V. Shamolin, “Jacobi integrability of problem of a spatial pendulum placed into over-running medium flow,” in: Modelling and Study of Systems, Sci. Conf., May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 143.

  124. M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 2, 65–68 (1997).

  125. M. V. Shamolin, “Spatial dynamics of a rigid body interacting with a medium. Workshop in Mechanics of Systems and Problems of Motion Control and Navigation,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 4, 174 (1997).

  126. M. V. Shamolin, “Spatial Poincaré topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997).

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  127. M. V. Shamolin, “Absolute and relative structural stability in spatial dynamics of a rigid body interacting with a medium,” in: Proc. of Int. Conf. “Mathematics in Industry,” ICIM–98, Taganrog, June 29 — July 3, 1998 [in Russian], Taganrog State Pedagog. Inst., Taganrog (1998), pp. 332–333.

  128. M. V. Shamolin, “Families of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 6, 29–37 (1998).

  129. M. V. Shamolin, “On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).

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  130. M. V. Shamolin, “Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 2, 178–189 (1999).

  131. M. V. Shamolin, “New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999).

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

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

  134. M. V. Shamolin, “A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000).

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  135. M. V. Shamolin, “Jacobi integrability in problem of four-dimensional rigid body motion in a resisting medium.” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000).

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  136. M. V. Shamolin, “Jacobi integrability of problem of four-dimensional body motion in a resisting medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, August 21–26, 2000 [in Russian], Vladimir, Vladimir State Univ. (2000), pp. 196–197.

  137. M. V. Shamolin, “Mathematical modeling of interaction of a rigid body with a medium and new cases of integrability,” in: CD-Proc. of ECCOMAS 2000, Barcelona, Spain, 11–14 September, Barcelona (2000).

  138. 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 Int. Conf. in Differential and Integral Equations, Odessa, September 12–14, 2000 [in Russian], AstroPrint, Odessa (2000), pp. 294–295.

  139. 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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  140. M. V. Shamolin, “On roughness of dissipative systems and relative roughness of variable dissipation systems. Abstracts of reports of the workshop in vector and tensor analysis named after P. K. Rashevskii,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 63 (2000).

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

  142. M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in overrunning medium flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001).

  143. M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,” Prikl. Mekh., 37, No. 6, 74–82 (2001).

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  144. M. V. Shamolin, “Integrability of a problem of four-dimensional rigid body in a resisting medium. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Fundam. Prikl. Mat., 7, No. 1, 309 (2001).

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

  146. M. V. Shamolin, “New Jacobi integrable cases in dynamics of two-, three-, and four-dimensional rigid body interacting with a medium,” in: Abstracts 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.

  147. M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 1, 189–193 (2001).

  148. M. V. Shamolin, “New integrable cases in dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, July 1–6, 2002 [in Russian], Vladimir State Univ., Vladimir (2002), pp. 142–144.

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

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  150. M. V. Shamolin, “Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002).

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  151. M. V. Shamolin, “Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003).

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  152. M. V. Shamolin, “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, No. 1, 919–975 (2003).

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  153. M. V. Shamolin, “Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004).

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  154. 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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  155. M. V. Shamolin, “Some cases of integrability in dynamics of a rigid body interacting with a resisting medium,” in: Abstracts of Reports of Int. Conf. on Differential Equations and Dynamical Systems, Suzdal’, July 5–10, 2004 [in Russian], Vladimir State Univ., Vladimir (2004), pp. 296–298.

  156. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2004).

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

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

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

  161. M. V. Shamolin, “On a certain integrable case of equations of dynamics in so(4) × ℝ4,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005).

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

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

  164. M. V. Shamolin, Model Problem of Body Motion in a Resisting Medium with Account for Dependence of Resistance Force on Angular Velocity, Sci. Rep. of Inst. of Mech., Moscow State University No. 4818 [in Russian], Institute of Mechanics, Moscow State University, Moscow (2006).

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

  166. M. V. Shamolin, “To problem on rigid body spatial drag in a resisting medium,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 3, 45–57 (2006).

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

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

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  169. 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 Int. Congress, St. Petersburg, June 4–8, 2007 [in Russian], St. Petersburg State Univ., St. Petersburg (2007), p. 178.

  170. 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 Int. Conf. “Analysis and Singularities” Devoted to the 70th Anniversary of V. I. Arnol’d, August 20–24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110–112.

  171. M. V. Shamolin, “Cases of complete integrability in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of All-Russian Conf. “Modern Problems of Continuous 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.

  172. M. V. Shamolin, “Cases of complete integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Abstracts of Reports of Int. Conf. “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.

  173. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in medium flow taking account of rotational derivatives of moments of its action force,” Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela, No. 3, 187–192 (2007).

  174. M. V. Shamolin, “Integrability in elementary functions of variable dissipation systems. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 38 (2007).

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  175. M. V. Shamolin, “Integrability of problem of four-dimensional rigid body motion in a resisting medium. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 21 (2007).

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  176. M. V. Shamolin, “Integrability of strongly nonconservative systems in transcendental elementary functions. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 40 (2007).

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

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  178. M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 27 (2007).

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  179. M. V. Shamolin, “On integrability in transcendental functions. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 34 (2007).

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  180. M. V. Shamolin, “On integrability of motion of four-dimensional body-pendulum situated in over-running medium flow. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Fundam. Direct., 23, 37 (2007).

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  181. 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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  182. M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2007).

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

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

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

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  186. M. V. Shamolin, “Integrability of some classes of dynamic systems in terms of elementary functions,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 43–49 (2008).

  187. M. V. Shamolin, “Methods of analysis of dynamic systems with various dissipation in dynamics of a rigid body,” in: CD-Proc. of ENOC-2008, St. Petersburg, Russia, June 30 — July 4, 2008 [in Russian].

  188. M. V. Shamolin, “New cases of complete integrability in dynamics of symmetric four-dimensional rigid body in nonconservative field,” in: Materials of Int. Conf. “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.

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

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  190. 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.

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

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  192. 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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  193. M. V. Shamolin, “Case of complete integrability in Dynamics of symmetric four-dimensional rigid body in a nonconservative field. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Its Appl., 65, 9 (2009).

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  194. M. V. Shamolin, “Cases of integrability of motion equations of four-dimensional rigid body in a nonconservative field of forces,” in: Materials of Int. Conf. “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], Universitet. Kniga, Moscow (2009), p. 233.

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

  196. M. V. Shamolin, “Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field,” Contemp. Math. Its Appl., 65, 132–142 (2009).

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  197. M. V. Shamolin, “Dynamical systems with variable dissipation: Methods and applications,” in: Proc. of 10th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2009 ), Lodz, Poland, December 7–10, 2009, Tech. Univ. Lodz (2009), pp. 91–104.

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

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

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  200. M. V. Shamolin, “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, 139–140 (2009).

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  201. M. V. Shamolin, “New cases of integrability in dynamics of four-dimensional rigid body in a nonconservative field. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Its Appl., 65, 6 (2009).

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  202. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” Contemp. Math. Its Appl., 62, 131–171 (2009).

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  203. M. V. Shamolin, “Stability of a rigid body translating in a resisting medium,” Prikl. Mekh., 45, No. 6, 125–140 (2009).

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

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

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

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  207. M. V. Shamolin, “Cases of complete integrability of spatial dynamics equations of a rigid body in a resisting medium,” in: Abstracts of Reports of Sci. Conf. “Lomonosov Readings,” Sec. Mechanics, April, 2010, Moscow, Lomonosov Moscow State Univ. [in Russian], MGU, Moscow (2010), p. 172.

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

  209. 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 Int. Conf. “Stability and Oscillations of Nonlinear Control Systems,” Moscow, IPU RAN, June 1–4, 2010 [in Russian], Moscow, IPU RAN (2010), pp. 429–431.

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

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

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  212. M. V. Shamolin, “Integrability and nonintegrability of dynamical systems in transcendental functions,” in: Abstracts of Reports of Voronezh Winter Math. School of S. G. Kreyn, Voronezh, 2010 [in Russian], Voronezh State Univ., Voronezh (2010), pp. 159–160.

  213. 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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  214. M. V. Shamolin, “On the problem of the motion of the body with front flat butt end in a resisting medium,” Sci. Rep. of Inst. of Mech., Moscow State University No. 5052 [in Russian], Institute of Mechanics, Moscow State University, Moscow (2010).

  215. 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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  216. M. V. Shamolin, “A multiparameter family of phase portraits in the dynamics of a rigid body interacting with a medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 24–30 (2011).

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

  219. 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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  220. M. V. Shamolin, “Complete lists of first integrals in dynamics of four-dimensional rigid body in a nonconservative force,” in: Abstracts of Reports of Int. Conf. Devoted to the 110th Anniversary of I. G. Petrovskii, 2011, Moscow [in Russian], MGU, Intuit, Moscow (2011), pp. 389–390.

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

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  222. M. V. Shamolin, “New case of complete integrability of the dynamic equations on the tangent stratification of three-dimensional sphere,” Vestn. SamGU. Estestvennonauch. Ser., No. 5 (86), 187–189 (2011).

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  223. M. V. Shamolin, “Rigid body motion in a resisting medium,” Mat. Model., 23, No. 12, 79–104 (2011).

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

  225. 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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  226. 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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  227. M. V. Shamolin, “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices,” Proc. Appl. Math. Mech., 12, 43–44 (2012).

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  228. M. V. Shamolin, “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices,” in: Book of Abstracts of 83rd Annual Sci. Conf. of the Int. Assoc. of Appl. Math. and Mech., Darmstadt, Germany, March 26–30, 2012, TU Darmstadt, Darmstadt (2012), p. 48.

  229. M. V. Shamolin, “Cases of complete integrability in transcendental functions in dynamics of a rigid body interacting with a medium. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,” Contemp. Math. Its Appl., 76, 7 (2012).

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

  231. M. V. Shamolin, “Cases of integrability in dynamics of a rigid body interacting with a resistant medium,” in: CD-Proc., 23th Int. Congress of Theoretical and Applied Mechanics, August 19–24, 2012, Beijing, China, China Sci. Lit. Publ. House, Beijing (2012).

  232. M. V. Shamolin, “Cases of integrability in dynamics of a rigid body interacting with a resistant medium,” in: Abstract Book, 23th Int. Congress of Theoretical and Applied Mechanics, August 19–24, 2012, Beijing, China, China Sci. Lit. Publ. House, Beijing (2012), p. 51.

  233. 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 Int. Sci. Conf. “Sixth Polyakhov Readings,” St. Petersburg, January 31 — February 3, 2012 [in Russian], I. V. Balabanov Publ., St. Petersburg (2012), p. 75.

  234. 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 Int. Chetaev Conf. “Analytical Mechanics, Stability and Control,” Kazan’, Russia, June 12–16, 2012 [in Russian], Kazan’ State Tech. Univ., Kazan’ (2012), pp. 508–514.

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    M. V. Shamolin

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 4, pp. 3–231, 2015.

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Shamolin, M.V. Integrable Variable Dissipation Systems on the Tangent Bundle of a Multi-Dimensional Sphere and Some Applications. J Math Sci 230, 185–353 (2018). https://doi.org/10.1007/s10958-018-3738-8

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