# Classification of Integrable Cases in the Dynamics of a Four-Dimensional Rigid Body in a Nonconservative Field in the Presence of a Tracking Force

## Abstract

This paper is a survey of integrable cases in the dynamics of a four-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. The problems examined are described by dynamical systems with so-called variable dissipation with zero mean.

The problem of a search for complete sets of transcendental first integrals of systems with dissipation is quite current; 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 a nontrivial symmetry group 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 results obtained, we analyze dynamical systems that appear in the dynamics of a four-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.

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### References

- 1.R. R. Aidagulov and M. V. Shamolin, “A phenomenological approach to the definition of interphase forces,”
*Dokl. Ross. Akad. Nauk*,**412**, No. 1, 44–47 (2007).MathSciNetGoogle Scholar - 2.R. R. Aidagulov and M. V. Shamolin, “Averaging operators and real equations of hydromechanics,”
*J. Math. Sci.*,**165**, No. 6, 637–653 (2010).MATHMathSciNetGoogle Scholar - 3.A. A. Andronov,
*Collection of Works*[in Russian], Izd. Akad. Nauk SSSR, Moscow (1956).Google Scholar - 4.A. A. Andronov and L. S. Pontryagin, “Rough systems,”
*Dokl. Akad. Nauk SSSR*,**14**, No. 5, 247–250 (1937).Google Scholar - 5.P. Appel,
*Theoretical Mechanics*, Vols. I, II [Russian translation], Fizmatgiz, Moscow (1960).Google Scholar - 6.V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, “Mathematical aspects of classical and celestial mechanics,” in:
*Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Direction*[in Russian], Vol. 3, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1985).Google Scholar - 7.A. V. Belyaev, “On the motion of a multi-dimensional body with a clamped point in the gravity force field,”
*Mat. Sb.*,**114**, No. 3, 465–470 (1981).MathSciNetGoogle Scholar - 8.I. Bendixson, “Sur les courbes définies par les équations différentielles,”
*Acta Math.*,**24**, 1–30 (1901).MathSciNetGoogle Scholar - 9.O. I. Bogoyavlenskii, “Some integrable cases of Euler equation,”
*Dokl. Akad. Nauk SSSR*,**287**, No. 5, 1105–1108 (1986).MathSciNetGoogle Scholar - 10.O. I. Bogoyavlenskii and G. F. Ivakh, “Topological analysis of integrable cases of V. A. Steklov,”
*Usp. Mat. Nauk*,**40**, No. 4, 145–146 (1985).MathSciNetGoogle Scholar - 11.A. D. Bryuno,
*Local Method of Nonlinear Analysis of Differential Equations*[in Russian], Nauka, Moscow (1979).Google Scholar - 12.N. Bourbaki,
*Integration*[Russian translation], Nauka, Moscow (1970).Google Scholar - 13.N. Bourbaki,
*Lie Groups and Lie Algebras*[Russian translation], Nauka, Moscow (1970).Google Scholar - 14.G. S. Byushgens and R. V. Studnev,
*Dynamics of Longitudinal and Lateral Motion*[in Russian], Mashinostroenie, Moscow (1969).Google Scholar - 15.G. S. Byushgens and R. V. Studnev,
*Dynamics of Aircrafts. Spatial Motion*[in Russian], Mashinostroenie, Moscow (1983).Google Scholar - 16.S. A. Chaplygin, “On the motion of heavy bodies in an incompressible fluid,” In:
*A Complete Collection of Works*[in Russian], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133–135.Google Scholar - 17.S. A. Chaplygin,
*Selected Works*[in Russian], Nauka, Moscow (1976).Google Scholar - 18.B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko,
*Modern Geometry. Theory and Applications*[in Russian], Nauka, Moscow (1979).Google Scholar - 19.D. V. Georgievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with a fixed point in ℝ
^{n},”*Dokl. Ross. Akad. Nauk,***380**, No. 1, 47–50 (2001).MathSciNetGoogle Scholar - 20.D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in ℝ
^{n},”*Dokl. Ross. Akad. Nauk,***383**, No. 5, 635–637 (2002).MathSciNetGoogle Scholar - 21.D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion for a generalized gyroscope in ℝ
^{n},”*Vestn. MGU, Ser. 1, Mat., Mekh.,***5**, 37–41 (2003).MathSciNetGoogle Scholar - 22.D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,”
*J. Math. Sci.*,**154**, No. 4, 449–461 (2008).MATHMathSciNetGoogle Scholar - 23.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University “Urgent problems of geometry and mechanics” named after V. V. Trofimov,”
*J. Math. Sci.*,**154**, No. 4, 462–495 (2008).MATHMathSciNetGoogle Scholar - 24.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University “Urgent problems of geometry and mechanics” named after V. V. Trofimov,”
*J. Math. Sci.*,**161**, No. 5, 603–614 (2009).MATHGoogle Scholar - 25.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, ‘Topical Problems of Geometry and Mechanics’ named after V. V. Trofimov,”
*J. Math. Sci.*,**165**, No. 6, 607–615 (2010).MATHGoogle Scholar - 26.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, ‘Urgent Problems of Geometry and Mechanics’ named after V. V. Trofimov,”
*J. Math. Sci.*,**187**, No. 3, 269–271 (2012).MATHMathSciNetGoogle Scholar - 27.D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies,”
*J. Math. Sci.*,**187**, No. 3, 280–299 (2012).MATHMathSciNetGoogle Scholar - 28.V. V. Golubev,
*Lectures on Analytic Theory of Differential Equations*[in Russian], Gostekhizdat, Moscow–Leningrad (1950).Google Scholar - 29.V. V. Golubev,
*Lectures on Integration of Equations of Motion of a Heavy Rigid Body About a Fixed Point*[in Russian], Gostekhizdat, Moscow–Leningrad (1953).Google Scholar - 30.M. I. Gurevich,
*Theory of Jets of an Ideal Liquid*[in Russian], Nauka, Moscow (1979).Google Scholar - 31.T. A. Ivanova, “On Euler equations in models of theoretical physics,”
*Mat. Zametki*,**52**, No. 2, 43–51 (1992).Google Scholar - 32.C. G. J. Jacobi,
*Forlesungen über Dynamik*, Druck und Verlag von G. Reimer, Berlin (1884).Google Scholar - 33.V. V. Kozlov,
*Qualitative Analysis Methods in Rigid Body Dynamics*[In Russian], MGU, Moscow (1980).Google Scholar - 34.V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,”
*Usp. Mat. Nauk*,**38**, No. 1, 3–67 (1983).Google Scholar - 35.G. Lamb,
*Hydrodynamics*[Russian translation], Fizmatgiz, Moscow (1947).Google Scholar - 36.B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov,
*Introduction to the Problem on the Motion of a Body in a Resisting Medium*[in Russian], Moscow State Univ., Moscow (1986).Google Scholar - 37.A. M. Lyapunov, “A new integrability case of equations of rigid body motion in a fluid,” In:
*A Collection of Works*[in Russian], Vol. I, Izd. Akad. Nauk SSSR, Moscow (1954), pp. 320–324.Google Scholar - 38.Yu. I. Manin, “Algebraic aspects of theory of nonlinear differential equations,” in:
*Progress in Science and Technology, Series on Contemporary Problems in Mathematics*[in Russian], Vol. 11, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (1978), pp. 5–112.Google Scholar - 39.Z. Nitecki,
*Differentiable Dynamics: An Introduction to the Orbit Structure of Diffeomorphisms*, MIT Press (1971).Google Scholar - 40.H. Poincaré,
*On Curves Defined by Differential Equations*[Russian translation], OGIZ, Moscow–Leningrad (1947).Google Scholar - 41.H. Poincaré, “New methods in celestial mechanics,” in:
*Selected Works*[Russian translation], Vols. 1, 2, Nauka, Moscow (1971–1972).Google Scholar - 42.V. A. Samsonov and M. V. Shamolin, “On the problem of body motion in a resisting medium,”
*Vestn. MGU, Mat., Mekh.*,**3**, 51–54 (1989).MathSciNetGoogle Scholar - 43.L. I. Sedov,
*Continuous Medium Mechanics*[in Russian], Vols. 1, 2, Nauka, Moscow (1983–1984).Google Scholar - 44.M. V. Shamolin, “On the problem of body motion in a medium with resistance,”
*Vestn. MGU, Ser. 1, Mat., Mekh.,***1**, 52–58 (1992).Google Scholar - 45.M. V. Shamolin, “Closed trajectories of different topological type in the problem of body motion in a medium with resistance,”
*Vestn. MGU, Ser. 1, Mat., Mekh.,***2**, 52–56 (1992).Google Scholar - 46.M. V. Shamolin, “Classification of phase portraits in problem of body motion in a resisting medium in the presence of a linear damping moment,”
*Prikl. Mat. Mekh.*,**57**, No. 4, 40–49 (1993).MathSciNetGoogle Scholar - 47.M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on plane,”
*Vestn. MGU, Ser. 1, Mat., Mekh.,***1**, 68–71 (1993).MathSciNetGoogle Scholar - 48.M. V. Shamolin, “Applications of Poincaré topographical system methods and comparison systems in some concrete systems of differential equations,”
*Vestn. MGU, Ser. 1, Mat., Mekh.*,**2**, 66–70 (1993).MathSciNetGoogle Scholar - 49.M. V. Shamolin, “A new two-parameter family of phase portraits in the problem of body motion in a medium,”
*Dokl. Ross. Akad. Nauk*,**337**, No. 5, 611–614 (1994).MathSciNetGoogle Scholar - 50.M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new two-parameter family of phase portraits,”
*Vestn. MGU, Ser. 1, Mat., Mekh.*,**4**, 57–69 (1996).MathSciNetGoogle Scholar - 51.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).MathSciNetGoogle Scholar - 52.M. V. Shamolin, “Definition of relative roughness and two-parameter family of phase portraits in rigid body dynamics,”
*Usp. Mat. Nauk*,**51**, No. 1, 175–176 (1996).MathSciNetGoogle Scholar - 53.M. V. Shamolin, “Periodic and Poisson stable trajectories in the problem of body motion in a resisting medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela*,**2**, 55–63 (1996).Google Scholar - 54.M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela*,**2**, 65–68 (1997).Google Scholar - 55.M. V. Shamolin, “Spatial Poincar´e topographical systems and comparison systems,”
*Usp. Mat. Nauk*,**52**, No. 3, 177–178 (1997).MathSciNetGoogle Scholar - 56.M. V. Shamolin, “Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in:
*Proc. WCSMO-2, Zakopane, Poland, May 26–30, 1997*, Zakopane, Poland (1997), pp. 387–392.Google Scholar - 57.M. V. Shamolin, “On integrability in transcendental functions,”
*Usp. Mat. Nauk*,**53**, No. 3, 209–210 (1998).MathSciNetGoogle Scholar - 58.M. V. Shamolin, “Family of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela*,**6**, 29–37 (1998).Google Scholar - 59.M. V. Shamolin, “Some classical problems in a three-dimensional dynamics of a rigid body interacting with a medium,” in:
*Proc. ICTACEM’98, Kharagpur, India, December 1–5, 1998*, Indian Inst. of Technology, Kharagpur (1998).Google Scholar - 60.M. V. Shamolin, “Structural stability in 3D dynamics of a rigid body,” in:
*Proc. WCSMO-3, Buffalo, New York, May 17–21, 1999*, Buffalo (1999).Google Scholar - 61.M. V. Shamolin, “Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in:
*Proc. Congr. “Nonlinear Analysis and Its Applications,” Moscow, Russia, September 1–5, 1998*[in Russian], Moscow (1999), pp. 497–508.Google Scholar - 62.M. V. Shamolin, “Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela,***2**, 178–189 (1999).Google Scholar - 63.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).Google Scholar - 64.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).MathSciNetGoogle Scholar - 65.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).MathSciNetGoogle Scholar - 66.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).Google Scholar - 67.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 Int. Math. School “Lyapunov Function Method and Its Application,” Crimea, Alushta, September 5–13, 2000*[in Russian], Simferopol’ (2000), p. 169.Google Scholar - 68.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. “Differential and Integral Equations,” Odessa, September 12–14, 2000*[in Russian], AstroPrint, Odessa (2000), pp. 294–295.Google Scholar - 69.M. V. Shamolin, “On limit sets of differential equations near singular points,”
*Usp. Mat. Nauk*,**55**, No. 3, 187–188 (2000).MathSciNetGoogle Scholar - 70.M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in:
*Proc. 16th IMACS World Congress, Lausanne, Switzerland, August 21–25, 2000*, EPFL, Lausanne (2000).Google Scholar - 71.M. V. Shamolin, “Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” in:
*Proc. ECCOMAS, Barcelona, Spain, September 11–14, 2000*, Barcelona (2000).Google Scholar - 72.M. V. Shamolin, “Integrability of a problem of a four-dimensional rigid body in a resisting medium,” in:
*Abstracts of Sessions of Workshop “current Problems of Geometry and Mechanics,” Fund. Prikl. Mat.,***7**, No. 1, 309 (2001).Google Scholar - 73.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.Google Scholar - 74.M. V. Shamolin, “New Jacobi integrable cases in dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” in:
*Abstracts of Reports of VIII All-Russian Congress “Theoretical and Applied Mechanics,” Perm’, August 23–29, 2001*[in Russian], Ural Department of Russian Academy of Science, Ekaterinburg (2001), pp. 599–600.Google Scholar - 75.M. V. Shamolin, “On stability of motion of a body twisted around its longitudinal axis in a resisting medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela***1**, 189–193 (2001).Google Scholar - 76.M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in overrunning medium flow,”
*Vestn. MGU, Ser. 1, Mat., Mekh.,***5**, 22–28 (2001).MathSciNetGoogle Scholar - 77.M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,”
*Prikl. Mekh.*,**37**, No. 6, 74–82 (2001).MATHMathSciNetGoogle Scholar - 78.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. “Differential Equations and Dynamical Systems,” Suzdal’, July 1–6, 2002*[in Russian], Vladimir State University, Vladimir (2002), pp. 142–144.Google Scholar - 79.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).MathSciNetGoogle Scholar - 80.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).MATHMathSciNetGoogle Scholar - 81.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).MATHMathSciNetGoogle Scholar - 82.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).MATHMathSciNetGoogle Scholar - 83.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 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.Google Scholar - 84.M. V. Shamolin, “On a certain integrable case in dynamics on
*so*(4)*×*ℝ^{4},” in:*Abstracts of Reports of All-Russian Conference “Differential Equations and Their Applications,” Samara, June 27–July 2, 2005*[in Russian], Univers-Grupp, Samara (2005), pp. 97–98.Google Scholar - 85.M. V. Shamolin, “On a certain integrable case of equations of dynamics in
*so*(4)*×*ℝ^{4},”*Usp. Mat. Nauk*,**60**, No. 6, 233–234 (2005).MathSciNetGoogle Scholar - 86.M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking into account rotational derivatives of force moment in angular velocity,”
*Dokl. Ross. Akad. Nauk*,**403**, No. 4, 482–485 (2005).MathSciNetGoogle Scholar - 87.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).MATHMathSciNetGoogle Scholar - 88.M. V. Shamolin, “Structural stable vector fields in rigid body dynamics,” in:
*Proc. 8th Conf. “Dynamical Systems: Theory and Applications,” Lodz, Poland, December 12–15, 2005*,**1**, Tech. Univ. Lodz, Lodz (2005), pp. 429–436.Google Scholar - 89.M. V. Shamolin, “On the problem of the motion of a rigid body in a resisting medium,”
*Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela*,**3**, 45–57 (2006).Google Scholar - 90.M. V. Shamolin, “On a case of complete integrability in four-dimensional rigid body dynamics,” in:
*Abstracts of Reports of International Conference “Differential Equations and Dynamical Systems,” Vladimir, July 10–15, 2006*[in Russian], Vladimir State University, Vladimir (2006), pp. 226–228.Google Scholar - 91.M. V. Shamolin,
*Methods for Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics*[in Russian], Ekzamen, Moscow (2007).Google Scholar - 92.M. V. Shamolin, “Some model problems of dynamics of a rigid body interacting with a medium,”
*Prikl. Mekh.*,**43**, No. 10, 49–67 (2007).MATHMathSciNetGoogle Scholar - 93.M. V. Shamolin, “New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” in:
*Abstracts of Sessions of Workshop “Current Problems of Geometry and Mechanics,” J. Math. Sci.*,**154**, No. 4, 462–495 (2008).Google Scholar - 94.M. V. Shamolin, “On integrability of motion of four-dimensional body-pendulum situated in over-running medium flow,” in:
*Abstracts of Sessions of Workshop “Current Problems of Geometry and Mechanics,” J. Math. Sci.*,**154**, No. 4, 462–495 (2008).Google Scholar - 95.M. V. Shamolin, “A case of complete integrability in dynamics on a tangent bundle of a twodimensional sphere,”
*Usp. Mat. Nauk*,**62**, No. 5, 169–170 (2007).MathSciNetGoogle Scholar - 96.M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in medium flow taking into account rotational derivatives of moment of its action force,”
*Izv. Ross Akad. Nauk, Mekh. Tverdogo Tela*,**3**, 187–192 (2007).Google Scholar - 97.M. V. Shamolin, “Case of complete integrability in dynamics of a four-dimensional rigid body in a nonconcervative force field,” in:
*Abstracts of Reports of Int. Congress “Nonlinear Dynamical Analysis-2007,” St. Petersburg, June 4–8, 2007*[in Russian], St. Petersburg State University, St. Petersburg (2007), p. 178.Google Scholar - 98.M. V. Shamolin, “Cases of complete integrability in dynamics of a four-dimensional rigid body in a nonconservative force field,” In:
*Abstracts of Reports of Int. Conf. “Analysis and Singularities” dedicated to the 70th Anniversary of V. I. Arnold, August 20–24, 2007, Moscow*[in Russian], MIAN, Moscow (2007), pp. 110–112.Google Scholar - 99.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).MathSciNetGoogle Scholar - 100.M. V. Shamolin, “4D rigid body and some cases of integrability,” in:
*Abstr. ICIAM07, Zürich, Switzerland, June 16–20, 2007*, ETH, Zürich (2007), pp. 311.Google Scholar - 101.M. V. Shamolin, “The cases of integrability in a 2D-, 3D- and 4D-rigid body,” in:
*Proc. Int. Conf. “Dynamical Methods and Mathematical Modelling,” Valladolid, Spain, September 18–22, 2007*, ETSII, Valladolid (2007), pp. 31.Google Scholar - 102.M. V. Shamolin, “Cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,”
*Proc. 9th Conf. “Dynamical Systems: Theory and Applications,” Lodz, Poland, December 17–20, 2007*,**1**, Tech. Univ. Lodz, Lodz (2007), pp. 415–422.Google Scholar - 103.M. V. Shamolin, “Dynamical systems with variable dissipation: approaches, methods, and applications,”
*J. Dynam. Sci.*,**162**, No. 6, 741–908 (2008).MathSciNetGoogle Scholar - 104.M. V. Shamolin, “New integrable cases in the dynamics of a body interacting with a medium taking into account the dependence of the moment of the resistance force on the angular velocity,”
*Prikl. Mat. Mekh.*,**72**, No. 2, 273–287 (2008).MATHMathSciNetGoogle Scholar - 105.M. V. Shamolin, “New integrable case in the dynamics of a four-dimensional rigid body in a nonconservative field,” in:
*Proc. Voronezh Spring Math. School “Pontryagin Readings–XIX,” Voronezh, May 2008*, Voronezh State Univ., Voronezh (2008), pp. 231–232.Google Scholar - 106.M. V. Shamolin, “New cases of complete integrability in the dynamics of a symmetric four-dimensional rigid body in a nonconservative field,” in:
*Proc. Int. Conf. “Contemporary Problems of Mathematics, Mechanics, and Informatics” dedicated to the 85th Anniversary of L. A. Tolokonnikov, Tula, November 17–21, 2008*, Grif, Tula (2008), pp. 317–320.Google Scholar - 107.M. V. Shamolin, “On the integrability in elementary functions of some classes of dynamical systems,”
*Vestn. Mosk. Univ., Ser. 1, Mat. Mekh.*,**3**, 43–49 (2008).MathSciNetGoogle Scholar - 108.M. V. Shamolin, “Three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,”
*Dokl. Ross. Akad. Nauk*,**418**, No. 1. 46–51 (2008).MathSciNetGoogle Scholar - 109.M. V. Shamolin, “Methods of analysis of dynamic systems with various dissipation in dynamics of a rigid body,”
*Proc. ENOC-2008, June 30–July 4, 2008*, St. Petersburg, Russia (2008).Google Scholar - 110.M. V. Shamolin, “Some methods of analysis of dynamical systems with various dissipation in dynamics of a rigid body,”
*PAMM*,**8**, 10137–10138 (2008).Google Scholar - 111.M. V. Shamolin, “The various cases of complete integrability in dynamics of a rigid body interacting with a medium,” in:
*CD-Proc. ECCOMAS Thematic Conf. “Multibody Dynamics,” Warsaw, Poland, June 29–July 2. 2009*, Polish Acad. Sci., Warsaw (2009).Google Scholar - 112.M. V. Shamolin, “On the integrability in elementary functions of some classes of nonconservative dynamical systems,”
*J. Math. Sci.*,**161**, No. 5, 734–778 (2009).MATHMathSciNetGoogle Scholar - 113.M. V. Shamolin, “New cases of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,” in:
*Proc. Semin. “Current Problems of Geometry and Mechanics,” J. Math. Sci.*,**165**, No. 6, 607–615 (2009).Google Scholar - 114.M. V. Shamolin, “Cases of the complete integrability in the dynamics of a symmetric four-dimensional rigid body in a nonconservative field,” in:
*Proc. Semin. “Current Problems of Geometry and Mechanics,” J. Math. Sci.*,**165**, No. 6, 607–615 (2009).Google Scholar - 115.M. V. Shamolin, “Classification of cases of complete integrability in the dynamics of a symmetric four-dimensional rigid body in a nonconservative field,”
*J. Math. Sci.*,**165**, No. 6, 743–754 (2009).MathSciNetGoogle Scholar - 116.M. V. Shamolin, “New cases of complete integrability in the dynamics of a dynamically symmetric four-dimensional rigid body in a nonconservative field,”
*Dokl. Ross. Akad. Nauk*,**425**, No. 3, 338–342 (2009).MathSciNetGoogle Scholar - 117.M. V. Shamolin, “Cases of integrability of the equations of motion of a four-dimensional rigid body in a nonconservative field,” in:
*Proc. Int. Conf. “Contemporary Problems of Mathematics, Mechanics, and Their Applications” dedicated to the 70th Anniversary of Prof. V. A. Sadovnichy, Moscow, March 30–April 2, 2009*, Univ. Kniga, Moscow (2009), p. 233.Google Scholar - 118.M. V. Shamolin, “Dynamical systems with variable dissipation: methods and applications,” in:
*Proc. 10th Conf. “Dynamical Systems: Theory and Applications,” Poland, Lodz, December 7–10, 2009*, Tech. Univ. Lodz, Lodz (2009), pp. 91–104.Google Scholar - 119.M. V. Shamolin, “New cases of integrability in dynamics of a rigid body with the cone form of its shape interacting with a medium,”
*PAMM*,**9**, 139–140 (2009).Google Scholar - 120.M. V. Shamolin, “Dynamical systems with various dissipation: background, methods, applications,” in:
*CD-Proc. XXXVIII Summer School-Conf. “Advanced Problems in Mechanics,” St. Petersburg (Repino), Russia, July 1–5, 2010*, IPME, St. Petersburg (2010), pp. 612–621.Google Scholar - 121.M. V. Shamolin, “Integrability and nonintegrability in terms of transcendental functions in dynamics of a rigid body,”
*PAMM*,**10**, 63–64 (2010).Google Scholar - 122.M. V. Shamolin, “New cases of the integrability in the spatial dynamics of a rigid body,”
*Dokl. Ross. Akad. Nauk*,**431**, No. 3, 339–343 (2010).MathSciNetGoogle Scholar - 123.M. V. Shamolin, “Spatial motion of a rigid body in a resisting medium,”
*Prikl. Mekh.*,**46**, No. 7, 120–133 (2010).MathSciNetGoogle Scholar - 124.M. V. Shamolin, “Cases of complete integrability of the equations of motion of a dynamically symmetric four-dimensional rigid body in a nonconservative field,” in:
*Proc. Int. Conf. “Differential Equations and Dynamical Systems,” Suzdal’, July 2–7, 2010*, Vladimir State Univ., Vladimir (2010), pp. 195.Google Scholar - 125.M. V. Shamolin, “A case of complete integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,”
*Usp. Mat. Nauk*,**65**, No. 1, 189–190 (2010).MathSciNetGoogle Scholar - 126.M. V. Shamolin, “Motion of a rigid body in a resisting medium,”
*Mat. Model.*,**23**, No. 12, 79–104 (2011).MATHMathSciNetGoogle Scholar - 127.M. V. Shamolin, “On a multi-parameter family of phase portraits in the dynamics of a rigid body interacting with a medium,”
*Vestn. Mosk. Univ., Ser. 1, Mat. Mekh.*,**3**, 24–30 (2011).MathSciNetGoogle Scholar - 128.M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,”
*Dokl. Ross. Akad. Nauk*,**437**, No. 2, 190–193 (2011).MathSciNetGoogle Scholar - 129.M. V. Shamolin, “A new case of complete integrability of dynamical equations on the tangent bundle of the three-dimensional sphere,”
*Vestn. Samar. State Univ., Estestvennonauch. Ser., Miscellanious*,**5**, 187–189 (2011).MathSciNetGoogle Scholar - 130.M. V. Shamolin, “Complete lists of first integrals in the dynamics of a four-dimensional rigid body in a nonconservative field,” in:
*Proc. Int. Conf. dedicated to the 110th birthday of Prof. I. G. Petrovsky*, Moscow State Univ., Moscow (2011), pp. 389–390.Google Scholar - 131.M. V. Shamolin, “Complete list of first integrals in the problem on the motion of a four-dimensional rigid body in a nonconservative field under a linear damping,”
*Dokl. Ross. Akad. Nauk*,**440**, No. 2, 187–190 (2011).MathSciNetGoogle Scholar - 132.M. V. Shamolin, “Comparison of complete integrability cases in dynamics of two-, three-, and four-dimensional rigid bodies in a nonconservative field,” in:
*Proc. XV Int. Conf. “Dynamical System Modelling and Stability Investigation,” May 25–27, 2011*, Kiev (2011), pp. 139.Google Scholar - 133.M. V. Shamolin, “Cases of complete integrability in transcendental functions in dynamics and certain invariant indices,” in:
*CD-Proc. 5th Int. Sci. Conf. “Physics and Control” (PHYSCON 2011), Leon, September 5–8, 2011*, Leon, Spain (2011).Google Scholar - 134.M. V. Shamolin, “Variety of cases of integrability in dynamics of a 2D-, 3D-, and 4D-rigid body interacting with a medium,” in:
*Proc. 11th Conf. “Dynamical Systems: Theory and Applications,” Lodz, Poland, December 5–8, 2011*, Tech. Univ. Lodz, Lodz (2011), pp. 11–24.Google Scholar - 135.M. V. Shamolin, “Variety of cases of integrability in dynamics of a 2D- and 3D-rigid body interacting with a medium,” in:
*CD-Proc. 8th ESMC 2012, Graz, Austria, July 9–13, 2012*, Graz (2012).Google Scholar - 136.M. V. Shamolin, “Cases of integrability in dynamics of a rigid body interacting with a resistant medium,”
*CD-Proc. 23th Int. Congr. “Theoretical and Applied Mechanics,” August 19–24, 2012, Beijing, China*, : China Science Literature Publishing House, Beijing (2012).Google Scholar - 137.M. V. Shamolin, “Problem on the motion of a body in a resisting medium taking into account the dependence of the moment of the resistance on the angular velocity,”
*Mat. Model.*,**24**, No. 10, 109–132 (2012).MATHMathSciNetGoogle Scholar - 138.M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,”
*J. Math. Sci.*,**187**, No. 3, 346–359 (2012).MATHMathSciNetGoogle Scholar - 139.M. V. Shamolin, “Some questions of qualitative theory in dynamics of systems with variable dissipation,”
*J. Math. Sci.*,**189**, No. 2, 314–323 (2013).MATHMathSciNetGoogle Scholar - 140.M. V. Shamolin, “A new case of integrability in the spatial dynamics of a rigid body interacting with a medium taking into account linear damping,”
*Dokl. Ross. Akad. Nauk*,**442**, No. 4, 479–481 (2012).MathSciNetGoogle Scholar - 141.M. V. Shamolin, “A new case of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field taking into account linear damping,”
*Dokl. Ross. Akad. Nauk*,**444**, No. 5, 506–509 (2012).MathSciNetGoogle Scholar - 142.M. V. Shamolin, “Cases of integrability in the dynamics of a four-dimensional rigid body in a nonconservative field,”
*Proc. Int. Conf. “Voronezh Winter Mat. School of G. G. Krein,” Voronezh, January 25–30, 2012*, Voronezh State Univ., Voronezh (2012), pp. 213–215.Google Scholar - 143.M. V. Shamolin, “Review of cases of integrability in the dynamics of lower- and higher-dimensional rigid bodies in nonconservative fields,” in:
*Proc. Int. Conf. “Differential Equations and Dynamical Systems,” Suzdal’, June 29–July 4, 2012*, Vladimir State Univ., Vladimir (2012), pp. 179–180.Google Scholar - 144.M. V. Shamolin, “Complete list of first integrals of dynamical equations of motion of a rigid body in a resisting medium taking into account linear damping,”
*Vestn. Mosk. Univ., Ser. 1, Mat. Mekh.*,**4**, 44–47 (2012).MathSciNetGoogle Scholar - 145.V. A. Steklov,
*On the Motion of a Rigid Body in a Fluid*[in Russian], Khar’kov (1893).Google Scholar - 146.G. K. Suslov,
*Theoretical Mechanics*[in Russian], Gostekhizdat, Moscow (1946).Google Scholar - 147.V. V. Trofimov, “Euler equations on finite-dimensional solvable Lie groups,”
*Izv. Akad. Nauk SSSR, Ser. Mat.*,**44**, No. 5, 1191–1199 (1980).MATHMathSciNetGoogle Scholar - 148.V. V. Trofimov and A. T. Fomenko, “ A methodology for constructing Hamiltonian flows on symmetric spaces and integrability of certain hydrodynamic systems,”
*Dokl. Akad. Nauk SSSR*,**254**, No. 6, 1349–1353 (1980).MathSciNetGoogle Scholar - 149.V. V. Trofimov and M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems,”
*J. Math. Sci.*,**180**, No. 4, 365–530 (2012).MathSciNetGoogle Scholar - 150.S. V. Vishik and S. F. Dolzhanskii, “Analogs of Euler–Poisson equations and magnetic electrodynamic related to Lie groups,”
*Dokl. Akad. Nauk SSSR*,**238**, No. 5, 1032–1035.Google Scholar - 151.E. T. Whittaker,
*A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. With an Introduction to the Problem of Three Bodies*, At the University Press, Cambridge (1960).MATHGoogle Scholar - 152.N. E. Zhukovsky, “On the fall of a light oblong body rotating about its longitudinal axis,” in:
*Complete Works*[in Russian], Vol. 5, Fizmatlit, Moscow (1937), pp. 72–80, 100–115.Google Scholar