# 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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## References

- 1.S. A. Agafonov, D. V. Georgievskii, and M. V. Shamolin, “Some actual problems of geometry and mechanics. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Fundam. Direct.*,**23**, 34 (2007).Google Scholar - 2.R. R. Aidagulov and M. V. Shamolin, “A certain improvement of Convey algorithm,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 3, 53–55 (2005).Google Scholar - 3.R. R. Aidagulov and M. V. Shamolin, “Archimedean uniform structure,”
*Contemp. Math. Fundam. Direct.*,**23**, 46–51 (2007).MATHGoogle Scholar - 4.R. R. Aidagulov and M. V. Shamolin, “General spectral approach to continuous medium dynamics,”
*Contemp. Math. Fundam. Direct.*,**23**, 52–70 (2007).Google Scholar - 5.R. R. Aidagulov and M. V. Shamolin, “Phenomenological approach to definition of interphase forces,”
*Dokl. Ross. Akad. Nauk*,**412**, No. 1, 44–47 (2007).Google Scholar - 6.R. R. Aidagulov and M. V. Shamolin, “Varieties of continuous structures,”
*Contemp. Math. Fundam. Direct.*,**23**, 71–86 (2007).MATHGoogle Scholar - 7.R. R. Aidagulov and M. V. Shamolin, “Averaging operators and real equations of hydromechanics,”
*Contemp. Math. Its Appl.*,**65**, 31–47 (2009).MATHGoogle Scholar - 8.R. R. Aidagulov and M. V. Shamolin, “Groups of colors,”
*Contemp. Math. Its Appl.*,**62**, 15–27 (2009).MATHGoogle Scholar - 9.R. R. Aidagulov and M. V. Shamolin, “Pseudodifferential operators in the theory of multiphase, multi-rate flows,”
*Contemp. Math. Its Appl.*,**65**, 11–30 (2009).MATHGoogle Scholar - 10.R. R. Aidagulov and M. V. Shamolin, “Integration formulas of tenth order and higher,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 4, 3–7 (2010).Google Scholar - 11.A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier,
*Qualitative Theory of Second-Order Dynamical Systems*[in Russian], Nauka, Moscow (1966).MATHGoogle Scholar - 12.A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier,
*Bifurcation Theory of Dynamical Systems on the Plane*[in Russian], Nauka, Moscow (1967).Google Scholar - 13.V. I. Arnol’d, “Hamiltonian property of Euler equations of rigid body dynamics in ideal fluid,”
*Usp. Mat. Nauk*,**24**, No. 3, 225–226 (1969).Google Scholar - 14.V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt,
*Mathematical Aspects of Classical and Celestial Mechanics*[in Russian], All-Union Inst. for Sci. and Tech. Inf., Moscow (1985).Google Scholar - 15.D. Arrowsmith and C. Place,
*Ordinary Differential Equations. Qualitative Theory with Applications*[Russian translation], Mir, Moscow (1986).MATHGoogle Scholar - 16.A. V. Belyaev, “On many-dimensional body motion with clumped point in gravity force field,”
*Mat. Sb.*,**114**, No. 3, 465–470 (1981).MathSciNetGoogle Scholar - 17.O. I. Bogoyavlenskii, “Some integrable cases of Euler equation,”
*Dokl. Akad. Nauk SSSR*,**287**, No. 5, 1105–1108 (1986).MathSciNetGoogle Scholar - 18.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).Google Scholar - 19.I. T. Borisenok and M. V. Shamolin, “Solution of differential diagnosis problem,”
*Fundam. Prikl. Mat.*,**5**, No. 3, 775–790 (1999).MathSciNetMATHGoogle Scholar - 20.I. T. Borisenok and M. V. Shamolin, “Solution of differential diagnosis problem by statistical trial method,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 1, 29–31 (2001).Google Scholar - 21.N. Bourbaki,
*Integration*[Russian translation], Nauka, Moscow (1970).Google Scholar - 22.N. Bourbaki,
*Lie Groups and Algebras*[Russian translation], Mir, Moscow (1972).Google Scholar - 23.G. S. Byushgens and R. V. Studnev,
*Dynamics of Longitudinal and Lateral Motion*[in Russian], Mashinostroenie, Moscow (1969).Google Scholar - 24.G. S. Byushgens and R. V. Studnev,
*Airplane Dynamics. A Spatial Motion*[in Russian], Mashinostroenie, Moscow (1988).Google Scholar - 25.S. A. Chaplygin, “On motion of heavy bodies in an incompressible fluid,” in:
*A Complete Collection of Works*, Vol. 1 [in Russian], Izd. Akad. Nauk SSSR, Leningrad (1933), pp. 133–135.Google Scholar - 26.S. A. Chaplygin,
*Selected Works*[in Russian], Nauka, Moscow (1976).Google Scholar - 27.B. A. Dubrovin and S. P. Novikov, “On Poisson brackets of hydrodynamic type,”
*Dokl. Akad. Nauk SSSR*,**279**, No. 2, 294–297 (1984).MathSciNetMATHGoogle Scholar - 28.B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko,
*Modern Geometry. Theory and Applications*[in Russian], Nauka, Moscow (1979).MATHGoogle Scholar - 29.V. A. Eroshin, “Plate reflection from ideal incompressible fluid surface,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 6, 99–104 (1970).Google Scholar - 30.V. A. Eroshin, “Submergence of a disk into a compressible fluid at an angle to a free surface,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 2, 142–144 (1983).Google Scholar - 31.V. A. Eroshin, “Experimental study of compression forces excited in an elastic cylinder under its entrance into water,” in:
*Applied Problems of Solidity and Plasticity*, Issue 46 [in Russian], Gor’k. Gos. Univ., Gor’kii (1990), pp. 54–59.Google Scholar - 32.V. A. Eroshin,
*Penetration of an Elastic Cylinder into Water with a High Speed*, Preprint No. 5, Institute of Mechanics, Moscow State University, Moscow (1991).Google Scholar - 33.V. A. Eroshin, “Experimental study of entrance of an elastic cylinder into water with high speed,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 5, 20–30 (1992).Google Scholar - 34.V. A. Eroshin, G. A. Konstantinov, N. I. Romanenkov, and Yu. L. Yakimov, “Experimental finding of pressure on a disk under its submergence into a compressible fluid at an angle to a free surface,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 2, 21–25 (1988).Google Scholar - 35.V. A. Eroshin, G. A. Konstantinov, N. I. Romanenkov, and Yu. L. Yakimov, “Experimental finding of hydrodynamic force moment under an asymmetric penetration of a disk into a compressible fluid,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 5, 88–94 (1990).Google Scholar - 36.V. A. Eroshin, A. V. Plyusnin, Yu. A. Sozonenko, and Yu. L. Yakimov, “On methodology for studying bend oscillations of an elastic cylinder under its entrance into water at an angle to a free surface,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 6, 164–167 (1989).Google Scholar - 37.V. A. Eroshin, V. A. Privalov, and V. A. Samsonov, “Two model problems of body motion in a resisting medium,” in:
*Collection of Scientific-Methodological Papers in Theoretical Mechanics*, Issue 18 [in Russian], Nauka, Moscow (1987), pp. 75–78.Google Scholar - 38.V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, “Mathematical modeling in problem of body drag in a medium under streamline flow around. Abstract of Chebyshev Readings,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 6, 17 (1995).Google Scholar - 39.V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, “Model problem of body drag in a resisting medium under streamline flow around,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Zhidk. Gaza*, No. 3, 23–27 (1995).Google Scholar - 40.R. R. Fakhrudinova and M. V. Shamolin, “On preservation of phase volume in ‘zero mean’ variable dissipation systems. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Fundam. Prikl. Mat.*,**7**, No. 1, 311 (2001).Google Scholar - 41.R. R. Fakhrudinova and M. V. Shamolin, “On preservation of phase volume in ‘zero mean’ variable dissipation systems. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Fundam. Direct.*,**23**, 22 (2007).Google Scholar - 42.D. V. Georgievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with fixed point in ℝ
^{n},”*Dokl. Ross. Akad. Nauk*,**380**, No. 1, 47–50 (2001).MathSciNetGoogle Scholar - 43.D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with fixed point in ℝ
^{n}. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”*Fundam. Prikl. Mat.*,**7**, No. 1, 315 (2001).Google Scholar - 44.D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with fixed point in ℝ
^{n},”*Dokl. Ross. Akad. Nauk*,**383**, No. 5, 635–637 (2002).Google Scholar - 45.D. V. Georgievskii and M. V. Shamolin, “First integrals of motion of equations of motion for a generalized gyroscope in ℝ
^{n},”*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 5, 37–41 (2003).Google Scholar - 46.D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,”
*Contemp. Math. Fundam. Direct.*,**23**, 5–6 (2007).MATHGoogle Scholar - 47.D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in ℝ
^{n},” In:*Abstracts of Sessions of Workshop ‘Actual Problems of Geometry and Mechanics,’ Contemporary Mathematics, Fundamental Directions*[in Russian],**23**(2007), p. 30.Google Scholar - 48.D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with a fixed point in ℝ
^{n}. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”*Contemp. Math. Fundam. Direct.*,**23**, 24–25 (2007).Google Scholar - 49.D. V. Georgievskii and M. V. Shamolin, “First integrals for equations of motion of a generalized gyroscope in
*n*-dimensional space. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”*Contemp. Math. Fundam. Direct.*,**23**, 31 (2007).Google Scholar - 50.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).MATHCrossRefGoogle Scholar - 51.D. V. Georgievskii and M. V. Shamolin, “Π-theorem of dimensionality theory. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Its Appl.*,**65**, 3 (2009).Google Scholar - 52.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University ‘Actual Problems of Geometry and Mechanics’ named after V. V. Trofimov,”
*Contemp. Math. Its Appl.*,**65**, 3–10 (2009).MATHGoogle Scholar - 53.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).MATHCrossRefGoogle Scholar - 54.D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies,”
*Contemp. Math. Its Appl.*,**76**, 22–39 (2012).MathSciNetMATHGoogle Scholar - 55.D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Its Appl.*,**76**, 9 (2012).MATHGoogle Scholar - 56.D. V. Georgievskii and M. V. Shamolin, “Sessions of the workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University ‘Actual Problems of Geometry and Mechanics’ named after V. V. Trofimov,”
*Contemp. Math. Its Appl.*,**76**, 3–10 (2012).MATHGoogle Scholar - 57.D. V. Georievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics: problems, approaches, and methods. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Fundam. Prikl. Mat.*,**7**, No. 1, 301 (2001).Google Scholar - 58.D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “On certain topological invariants of flows with complex potential. Abstracts of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Fundam. Prikl. Mat.*,**7**, No. 1, 305 (2001).Google Scholar - 59.D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics: problems, approaches, and methods. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Fundam. Direct.*,**23**, 16 (2007).Google Scholar - 60.D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “On certain topological invariants of flows with complex potential. Abstract of sessions of the workshop ‘Actual Problems of Geometry and Mechanics’,”
*Contemp. Math. Fundam. Direct.*,**23**, 19 (2007).Google Scholar - 61.C. Godbillon,
*Differential Geometry and Analytical Mechanics*[Russian translation], Mir, Moscow (1973).Google Scholar - 62.V. V. Golubev,
*Lectures on Analytical Theory of Differential Equations*[in Russian], Gostekhizdat, Moscow (1950).Google Scholar - 63.V. V. Golubev,
*Lectures on Integrating Equations of Heavy Body Motion Around a Fixed Point*[in Russian], Gostekhizdat, Moscow (1953).MATHGoogle Scholar - 64.M. I. Gurevich,
*Jet Theory of Ideal Fluid*[in Russian], Nauka, Moscow (1979).Google Scholar - 65.Ph. Hartman,
*Ordinary Differential Equations*[Russian translation], Mir, Moscow (1970).Google Scholar - 66.T. A. Ivanova, “On Euler equations in models of theoretical physics,”
*Mat. Zametki*,**52**, No. 2, 43–51 (1992).MathSciNetMATHGoogle Scholar - 67.C. Jacobi,
*Lectures on Dynamics*[Russian translation], ONTI, Moscow (1936).Google Scholar - 68.V. V. Kozlov,
*Qualitative Analysis Methods in Rigid Body Dynamics*[In Russian], MGU, Moscow (1980).MATHGoogle Scholar - 69.V. V. Kozlov, “Hydrodynamics of Hamiltonian systems,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 6, 10–22 (1983).Google Scholar - 70.V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,”
*Usp. Mat. Nauk*,**38**, No. 1, 3–67 (1983).MathSciNetGoogle Scholar - 71.V. V. Kozlov, “To problem of rigid body rotation in magnetic field,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela*, No. 6, 28–33 (1985).Google Scholar - 72.V. V. Kozlov, “On rigid body fall in ideal fluid,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela*, No. 5, 10–17 (1989).Google Scholar - 73.V. V. Kozlov, “To problem of heavy rigid body fall in a resisting medium,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 1, 79–87 (1990).Google Scholar - 74.V. V. Kozlov and N. N. Kolesnikov, “On integrability of Hamiltonian systems,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 6, 88–91 (1979).Google Scholar - 75.V. V. Kozlov and D. A. Onishchenko, “Nonintegrability of Kirchhoff equations,”
*Dokl. Akad. Nauk SSSR*,**266**, No. 6, 1298–1300 (1982).MathSciNetMATHGoogle Scholar - 76.G. Lamb,
*Hydrodynamics*[Russian translation], Fizmatgiz, Moscow (1947).Google Scholar - 77.B. Ya. Lokshin, Yu. M. Okunev, V. A. Samsonov, and M. V. Shamolin, “Some integrable cases of rigid body spatial oscillations in a resisting medium,” in:
*Abstracts of Reports of XXI Scientific Readings in Cosmonautics, Moscow, January 28–31, 1997*[in Russian], Institute of History of Natural Sciences and Technics, Russian Academy of Sciences, Moscow (1997), pp. 82–83.Google Scholar - 78.B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov,
*An Introduction to Problem of Motion of a Body in a Resisting Medium*[in Russian], MGU, Moscow (1986).Google Scholar - 79.A. M. Lyapunov, “A new integrability case of equations of rigid body motion in a fluid,” in:
*A Collection of Works*, Vol. I [in Russian], Izd. Akad. Nauk SSSR, Moscow (1954), pp. 320–324.Google Scholar - 80.Z. Nitetski,
*Introduction to Differential Dynamics*[Russian translation], Mir, Moscow (1975).Google Scholar - 81.Yu. M. Okunev and M. V. Shamolin, “On integrability in elementary functions of certain classes of complex nonautonomous equations,”
*Contemp. Math. Its Appl.*,**65**, 122–131 (2009).MATHGoogle Scholar - 82.J. Palis and W. De Melu,
*Geometric Theory of Dynamical Systems. An Introduction*[Russian translation], Mir, Moscow (1986).Google Scholar - 83.M. Peixoto, “On structural stability,”
*Ann. Math.*,**2**, No. 69, 199–222 (1959).MATHCrossRefGoogle Scholar - 84.M. Peixoto, “Structural stability on two-dimensional manifolds,”
*Topology*,**1**, No. 2, p. 101–120 (1962).MathSciNetMATHCrossRefGoogle Scholar - 85.M. Peixoto, “On an approximation theorem of Kupka and Smale,”
*J. Differ. Equ.*,**3**, 214–227 (1966).MathSciNetMATHCrossRefGoogle Scholar - 86.H. Poincaré,
*On Curves Defined by Differential Equations*[Russian translation], OGIZ, Moscow (1947).Google Scholar - 87.H. Poincaré,
*New Methods in Celestial Mechanics*, in:*Selected Works*, Vols. 1, 2 [Russian translation], Nauka, Moscow (1971–1972).Google Scholar - 88.H. Poincaré,
*On Science*[Russian translation], Nauka, Moscow (1983).Google Scholar - 89.N. V. Pokhodnya and M. V. Shamolin, “New case of integrability in dynamics of multi-dimensional body,”
*Vestn. SamGU. Estestvennonauch. Ser.*, No. 9 (100), 136–150 (2012).Google Scholar - 90.N. V. Pokhodnya and M. V. Shamolin, “Certain conditions of integrability of dynamical systems in transcendental functions,”
*Vestn. SamGU. Estestvennonauch. Ser.*, No. 9/1 (110), 35–41 (2013).Google Scholar - 91.N. L. Polyakov and M. V. Shamolin, “On closed symmetric class of function preserving every unary predicate,”
*Vestn. SamGU. Estestvennonauch. Ser.*, No. 6 (107), 61–73 (2013).Google Scholar - 92.N. L. Polyakov and M. V. Shamolin, “On a generalization of Arrow’s impossibility theorem,”
*Dokl. Ross. Akad. Nauk*,**456**, No. 2, 143–145 (2014).MATHGoogle Scholar - 93.L. Prandtl and A. Betz,
*Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen*, Berlin (1932).Google Scholar - 94.V. A. Samsonov and M. V. Shamolin, “On body motion in a resisting medium,” in:
*Contemporary Problems of Mechanics and Technologies of Machine Industry, All-Union Conf., April, 16–18, 1989. Abstracts of Reports*[in Russian], All-Union Institute for Scientific and Technical Information, Moscow (1989), pp. 128–129.Google Scholar - 95.V. A. Samsonov and M. V. Shamolin, “To problem on body motion in a resisting medium,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 3, 51–54 (1989).Google Scholar - 96.V. A. Samsonov and M. V. Shamolin,
*A Model Problem of Body Motion in a Medium with Streamline Flow around*, Sci. Rep. of Inst. of Mech., Moscow State University No. 3969 [in Russian], Institute of Mechanics, Moscow State University, Moscow (1990).Google Scholar - 97.V. A. Samsonov andM. V. Shamolin, “A model problem of body motion in a medium with streamline flow around,” in:
*Nonlinear Oscillations of Mechanical Systems. Abstract of Reports of II All-Union Conf., September, 1990*, Pt. 2 [in Russian], Gor’kii (1990), pp. 95–96.Google Scholar - 98.V. A. Samsonov and M. V. Shamolin,
*To Problem of Body Drag in a Medium under Streamline Flow around*, Sci. Rep. of Inst. of Mech., Moscow State University No. 4141 [in Russian], Institute of Mechanics, Moscow State University, Moscow (1991).Google Scholar - 99.V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin,
*Mathematical Modelling in Problem of Body Drag in a Resisting Medium under Streamline Flow around*, Sci. Rep. of Inst. of Mech., Moscow State University No. 4396 [in Russian], Institute of Mechanics, Moscow State University, Moscow (1995).Google Scholar - 100.L. I. Sedov,
*Continuous Medium Mechanics*, Vols. 1, 2 [in Russian], Nauka, Moscow (1983–1984).Google Scholar - 101.N. Yu. Selivanova and M. V. Shamolin, “Studying the interphase zone in a certain singular-limit problem,” in:
*Materials of Voronezh All-Russian Conf. ‘Pontryagin Readings-XXII’, Voronezh, May 3–9, 2011*[in Russian], Voronezh State University, Voronezh (2011), pp. 164–165.Google Scholar - 102.N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a certain problem with free boundary,”
*Vestn. SamGU. Estestvennonauch. Ser.*, No. 8 (89), 86–94 (2011).Google Scholar - 103.N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a one-phase problem with free boundary,” in:
*Materials of Voronezh Winter Math. School “Contemporary Methods of Function Theory and Adjacent Problems,” Voronezh, January 26 — February 1, 2011*[in Russian], Voronezh State University, Voronezh (2011), p. 307.Google Scholar - 104.N. Yu. Selivanova and M. V. Shamolin, “Local solvability of a one-phase problem with free boundary,”
*Contemp. Math. Its Appl.*,**78**, 99–108 (2012).MATHGoogle Scholar - 105.N. Yu. Selivanova and M. V. Shamolin, “Local solvability of the capillary problem,”
*Contemp. Math. Its Appl.*,**78**, 119–125 (2012).MATHGoogle Scholar - 106.N. Yu. Selivanova and M. V. Shamolin, “Quasi-stationary Stefan problem with values on front depending on its geometry,”
*Contemp. Math. Its Appl.*,**78**, 126–134 (2012).MATHGoogle Scholar - 107.N. Yu. Selivanova and M. V. Shamolin, “Studying the interphase zone in a certain singular-limit problem,”
*Contemp. Math. Its Appl.*,**78**, 109–118 (2012).MATHGoogle Scholar - 108.M. V. Shamolin, “Closed trajectories of different topological type in problem of body motion in a medium with resistance,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 2, 52–56 (1992).Google Scholar - 109.M. V. Shamolin, “To problem of body motion in a medium with resistance,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 1, 52–58 (1992).Google Scholar - 110.M. V. Shamolin, “Applications of Poincar´e topographical system methods and comparison systems in some concrete systems of differential equations,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 2, 66–70 (1993).Google Scholar - 111.M. V. Shamolin, “Phase pattern classification for the problem of the motion of a body in a resisting medium in the presence of a linear damping moment,”
*Prikl. Mat. Mekh.,***57**, No. 4, 40–49 (1993).MathSciNetMATHGoogle Scholar - 112.M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely distant points as limit sets for dynamical systems on plane,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 1, 68–71 (1993).Google Scholar - 113.M. V. Shamolin, “A new two-parameter family of phase portraits in problem of a body motion in a medium,”
*Dokl. Ross. Akad. Nauk*,**337**, No. 5, 611–614 (1994).Google Scholar - 114.M. V. Shamolin, “On relative stability of dynamical systems in problem of body motion in a resisting medium. Abstracts of Reports of Chebyshev Readings,”
*Vestn. VGU. Ser. 1 Mat., Mekh.*,**6**, 17 (1995).Google Scholar - 115.M. V. Shamolin, “Relative structural stability of dynamical systems for problem of body motion in a medium,” in:
*Analytical, Numerical, and Experimental Methods in Mechanics. A Collection of Scientific Works*[in Russian], MGU, Moscow (1995), pp. 14–19.Google Scholar - 116.M. V. Shamolin, “A list of integrals of dynamical equations in spatial problem of body motion in a resisting medium,” in:
*Modelling and Study of Stability of Systems. Sci. Conf. May 20–24, 1996. Abstracts of Reports (Study of Systems)*[in Russian], Kiev (1996), p. 142.Google Scholar - 117.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).MATHCrossRefGoogle Scholar - 118.M. V. Shamolin, “Introduction to problem of body drag in a resisting medium and a new two-parameter family of phase portraits,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 4, 57–69 (1996).Google Scholar - 119.M. V. Shamolin, “Introduction to spatial dynamics of rigid body motion in resisting medium.” in:
*Materials of Int. Conf. and Chebyshev Readings Devoted to the 175th Anniversary of P. L. Chebyshev, Moscow, May 14–19, 1996*, Vol. 2 [in Russian], MGU, Moscow (1996), pp. 371–373.Google Scholar - 120.M. V. Shamolin, “On a certain integrable case in dynamics of spatial body motion in a resisting medium,” in:
*II Symp. in Classical and Celestial Mechanics. Abstracts of Reports. Velikie Luki, August 23–28, 1996*[in Russian], Velikie Luki (1996), pp. 91–92.Google Scholar - 121.M. V. Shamolin, “Periodic and Poisson stable trajectories in problem of body motion in a resisting medium,”
*Izv. Akad. Nauk SSSR. Ser. Mekh. Tverd. Tela*, No. 2, 55–63 (1996).Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).Google Scholar - 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).Google Scholar - 126.M. V. Shamolin, “Spatial Poincaré topographical systems and comparison systems,”
*Usp. Mat. Nauk*,**52**, No. 3, 177–178 (1997).MATHCrossRefGoogle Scholar - 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.Google Scholar - 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).Google Scholar - 129.M. V. Shamolin, “On integrability in transcendental functions,”
*Usp. Mat. Nauk*,**53**, No. 3, 209–210 (1998).MathSciNetMATHCrossRefGoogle Scholar - 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).Google Scholar - 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).Google Scholar - 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).CrossRefGoogle Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 139.M. V. Shamolin, “On limit sets of differential equations near singular points,”
*Usp. Mat. Nauk*,**55**, No. 3, 187–188 (2000).MathSciNetMATHCrossRefGoogle Scholar - 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).Google Scholar - 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.Google Scholar - 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).Google Scholar - 143.M. V. Shamolin, “Integrability cases of equations for spatial dynamics of a rigid body,”
*Prikl. Mekh.,***37**, No. 6, 74–82 (2001).MathSciNetMATHGoogle Scholar - 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).MathSciNetGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 149.M. V. Shamolin, “On integrability of certain classes of nonconservative systems,”
*Usp. Mat. Nauk*,**57**, No. 1, 169–170 (2002).CrossRefGoogle Scholar - 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).MathSciNetMATHCrossRefGoogle Scholar - 151.M. V. Shamolin, “Foundations of differential and topological diagnostics,”
*J. Math. Sci.*,**114**, No. 1, 976–1024 (2003).MathSciNetMATHCrossRefGoogle Scholar - 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).MathSciNetMATHCrossRefGoogle Scholar - 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).MathSciNetMATHCrossRefGoogle Scholar - 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).MathSciNetMATHGoogle Scholar - 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.Google Scholar - 156.M. V. Shamolin,
*Some Problems of Differential and Topological Diagnosis*[in Russian], Ekzamen, Moscow (2004).Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).MathSciNetMATHGoogle Scholar - 160.M. V. Shamolin, “On a certain integrable case in dynamics on so(4)
*×*ℝ^{4},” 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.Google Scholar - 161.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).MathSciNetCrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).MathSciNetMATHCrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 177.M. V. Shamolin,
*Methods for Analysis Variable Dissipation Dynamical Systems in Rigid Body Dynamics*[in Russian], Ekzamen, Moscow (2007).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).MathSciNetMATHGoogle Scholar - 182.M. V. Shamolin,
*Some Problems of Differential and Topological Diagnosis*[in Russian], Ekzamen, Moscow (2007).Google Scholar - 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.Google Scholar - 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.Google Scholar - 185.M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications,”
*Fundam. Prikl. Mat.*,**14**, No. 3, 3–237 (2008).MathSciNetGoogle Scholar - 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).Google Scholar - 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].Google Scholar - 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.Google Scholar - 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).MathSciNetMATHGoogle Scholar - 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.Google Scholar - 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).CrossRefGoogle Scholar - 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).MathSciNetMATHGoogle Scholar - 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).Google Scholar - 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.Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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.Google Scholar - 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).MATHGoogle Scholar - 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).CrossRefGoogle Scholar - 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).Google Scholar - 202.M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,”
*Contemp. Math. Its Appl.*,**62**, 131–171 (2009).MATHGoogle Scholar - 203.M. V. Shamolin, “Stability of a rigid body translating in a resisting medium,”
*Prikl. Mekh.*,**45**, No. 6, 125–140 (2009).MathSciNetMATHGoogle Scholar - 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.Google Scholar - 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).Google Scholar - 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).CrossRefGoogle Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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).CrossRefGoogle Scholar - 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.Google Scholar - 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).MATHGoogle Scholar - 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).Google Scholar
- 215.M. V. Shamolin, “Spatial motion of a rigid body in a resisting medium,”
*Prikl. Mekh.*,**46**, No. 7, 120–133 (2010).MathSciNetGoogle Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).Google Scholar - 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.Google Scholar - 221.M. V. Shamolin, “Dynamical invariants of integrable variable dissipation dynamical systems,”
*Vestn. Nizhegorod. Univ.*,**2**, No. 4, 356–357 (2011).Google Scholar - 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).Google Scholar - 223.M. V. Shamolin, “Rigid body motion in a resisting medium,”
*Mat. Model.*,**23**, No. 12, 79–104 (2011).MathSciNetMATHGoogle Scholar - 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.Google Scholar - 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).Google Scholar - 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).Google Scholar - 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).CrossRefGoogle Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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).Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 235.M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,”
*Contemp. Math. Its Appl.*,**76**, 84–99 (2012).MathSciNetMATHGoogle Scholar - 236.M. V. Shamolin, “Complete list of first integrals of dynamic equations of spatial rigid body motion in a resisting medium under assumption of linear damping,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 4, 43–47 (2012).Google Scholar - 237.M. V. Shamolin, “New cases of integrability in transcendental functions in rigid body dynamics in a nonconservative field,” in:
*Materials of Voronezh Spring Math. School “Pontryagin Readings-XXIII,” Voronezh, May 3–9, 2012*[in Russian], Voronezh State Univ., Voronezh (2012), p. 200.Google Scholar - 238.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 Int. Conf. “Stability and Oscillations of Nonlinear Control Systems,” Moscow, IPU RAN, June 5–8, 2012*[in Russian], Moscow, IPU RAN (2012), pp. 339–341.Google Scholar - 239.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 Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, June 26 — July 4, 2012*[in Russian], Suzdal’ (2012), pp. 179–180.Google Scholar - 240.M. V. Shamolin, “Some questions of qualitative theory in dynamics of systems with the variable dissipation,”
*Contemp. Math. Its Appl.*,**78**, 138–147 (2012).Google Scholar - 241.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,”
*Mat. Model.*,**24**, No. 10, 109–132 (2012).MathSciNetMATHGoogle Scholar - 242.M. V. Shamolin, “Variety of the cases of integrability in dynamics of a 2D-, and 3D-rigid body interacting with a medium,” in:
*8th ESMC 2012, CD-Materials*(*Graz, Austria, July 9–13, 2012*)*, Graz*, Graz, Austria (2012).Google Scholar - 243.M. V. Shamolin, “A new case of integrability in transcendental functions in the dynamics of solid body interacting with the environment,”
*Avtomat. Telemekh.*, No. 8, 173–190 (2013).Google Scholar - 244.M. V. Shamolin, “Cases of integrability in transcendental functions in 3D dynamics of a rigid body interacting with a medium,” in:
*Proc. ECCOMAS Multibody Dynamics 2013, 1–4 July, 2013, Univ. of Zagreb, Croatia*, Univ. of Zagreb (2013), pp. 903–912.Google Scholar - 245.M. V. Shamolin, “Complete list of first integrals of dynamic equations of motion of a 4D rigid body in a nonconservative field under the assumption of linear damping,”
*Dokl. Ross. Akad. Nauk*,**449**, No. 4, 416–419 (2013).Google Scholar - 246.M. V. Shamolin, “Dynamical pendulum-like nonconservative systems,” in:
*12th Conf. on Dynamical Systems*(*Theory and Applications*) (*DSTA 2013*)*, Abstracts, Lodz, Poland, December 2–5, 2013*, Tech. Univ. Lodz (2013), pp. 160.Google Scholar - 247.M. V. Shamolin, “New case of integrability in the dynamics of a multidimensional solid in a nonconservative field,”
*Dokl. Ross. Akad. Nauk*,**453**, No. 1, 46–49 (2013).Google Scholar - 248.M. V. Shamolin, “New case of integrability of dynamic equations on the tangent bundle of a 3-sphere,”
*Usp. Mat. Nauk*,**68**, No. 5, 185–186 (2013).MathSciNetMATHCrossRefGoogle Scholar - 249.M. V. Shamolin, “On integrability in dynamic problems for a rigid body interacting with a medium,”
*Prikl. Mekh.*,**49**, No. 6, 44–54 (2013).MathSciNetMATHGoogle Scholar - 250.M. V. Shamolin, “Review of cases of integrability in dynamics of low- and multidimensional rigid body in a nonconservative field,” in:
*XXXIII Int. Conf. Dynamics Days Europe 2013, June 3–7, 2013, Madrid, Spain, Book of Abstracts*, CTB UPM, Madrid (2013), p. 157, http://www.dynamics-days-europe-2013.org/DDEXXXIII-AbstractsBook.pdf. - 251.M. V. Shamolin, “Review of integrable cases in dynamics of low- and multidimensional rigid body in a nonconservative field,” in:
*Advanced Problems in Mechanics: Book of Abstracts of Int. Summer School-Conf., July 1–6, 2013, St. Petersburg*, St. Petersburg, Polytech. Univ. Publ. House (2013), p. 99.Google Scholar - 252.M. V. Shamolin, “Variety of cases of integrability in dynamics of low- and many-dimensional rigid body in a nonconservative field of forces,”
*Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr.*,**125**, 5–254 (2013).Google Scholar - 253.M. V. Shamolin, “Variety of the cases of integrability in dynamics of a symmetric 2D-, 3D-, and 4D-rigid body in a nonconservative field,” in:
*Int. J. Struct. Stabil. Dynam.*,**13**, No. 7, 1340011 (2013).Google Scholar - 254.M. V. Shamolin, “A new case of integrability in the dynamics of a multidimensional solid in a nonconservative field under the assumption of linear damping,”
*Dokl. Ross. Akad. Nauk*,**457**, No. 5, 542–545 (2014).Google Scholar - 255.M. V. Shamolin, “Integrable various dissipation systems on tangent bundle of finite-dimensional sphere,” in:
*Materials of II Int. Conf. “Geometrical Analysis and its Applications,” Volgograd, May 26–30, 2014*[in Russian], VolGU, Volgograd (2014), pp. 143–145.Google Scholar - 256.M. V. Shamolin, “Problem on a body motion in a resisting medium under the action of a tracking force: qualitative analysis and integrability,” in:
*Proc. of XII All-Russian Counsel on Problems of Control*(*VSPU-2014*)*, Moscow, June 16–19, 2014*[in Russian], Electronic Source, Moscow, IPU RAN (2014), pp. 1813–1824.Google Scholar - 257.M. V. Shamolin, “Review of cases of integrability in dynamics of lower- and multidimensional rigid body in a nonconservative field of forces,” in:
*Recent Advances in Mathematics, Statistics and Economics. Proc. of 2014 Int. EUROPMENT Conf. on Pure Math.—Appl. Math.*(*PM-AM’14*)*, Venice, Italy, March 15–17, 2014*, Venice (2014), pp. 86–102.Google Scholar - 258.M. V. Shamolin, “Review of integrable cases of multidimensional rigid body motion equations in a nonconservative field,” in:
*Materials of Voronezh Winter Math. School of S. G. Kreyn, Voronezh, January, 2014*[in Russian], Voronezh State Univ., Voronezh (2014), pp. 404–408.Google Scholar - 259.M. V. Shamolin, “Integrable systems on tangent bundle of multidimensional sphere,”
*Tr. Semin. Petrovskogo*, to appear.Google Scholar - 260.M. V. Shamolin and S. V. Tsyptsyn,
*Analytical and Numerical Study of Trajectories of Body Motion in a Resisting medium*, Sci. Rep. of Inst. of Mech., Moscow State University No. 4289 [in Russian], Institute of Mechanics, Moscow State University, Moscow (1993).Google Scholar - 261.G. K. Suslov,
*Theoretical Mechanics*[in Russian], Gostekhizdat, Moscow (1946).Google Scholar - 262.E. I. Suvorova and M. V. Shamolin, “Poincaré topographical systems and comparison systems of higher orders,” in:
*Math. Conf. “Contemporary Methods of Function Theory and Related Problems,” Voronezh, January 26–February 2, 2003*[in Russian], Voronezh State University, Voronezh (2003), pp. 251–252.Google Scholar - 263.V. G. Tabachnikov, “Stationary characteristics of wings in small velocities under whole range of angles of attack,” in:
*Proc. of Central Aero-Hydrodynamical Inst.*, Issue 1621 [in Russian], Moscow (1974), pp. 18–24.Google Scholar - 264.V. V. Trofimov, “Embeddings of finite groups in compact Lie groups by regular elements,”
*Dokl. Akad. Nauk SSSR*,**226**, No. 4, 785–786 (1976).MathSciNetGoogle Scholar - 265.V. V. Trofimov, “Euler equations on finite-dimensional solvable Lie groups,”
*Izv. Akad. Nauk SSSR. Ser. Mat.*,**44**, No. 5, 1191–1199 (1980).MathSciNetMATHGoogle Scholar - 266.V. V. Trofimov, “Symplectic structures on automorphism groups of symmetric spaces,”
*Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.*, No. 6, 31–33 (1984).Google Scholar - 267.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 - 268.V. V. Trofimov and M. V. Shamolin, “Dissipative systems with nontrivial generalized Arnol’d–Maslov classes. 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, 62 (2000).Google Scholar - 269.V. V. Trofimov and M. V. Shamolin, “Geometrical and dynamical invariants of integrable Hamiltonian and dissipative systems,”
*Fundam. Prikl. Mat.*,**16**, No. 4, 3–229 (2010).Google Scholar - 270.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 (1978).MathSciNetGoogle Scholar - 271.Yu. G. Vyshkvarko and M. V. Shamolin, “Some problems of qualitative theory in rigid body dynamics”, in:
*All-Russian Conf. in Honor of the 110th Anniversary of Mathematics Faculty of MPSU “Mathematics, Informatics and Methodology of Its Teaching,” Moscow, March 14–16*[in Russian], Moscow, MPSU, pp. 40–41 (2011).Google Scholar - 272.C. Weyher, “Le vol plané,”
*Aéronaute*(1890).Google Scholar - 273.N. E. Zhukovskii, “On a fall of light oblong bodies rotating around their longitudinal axis,” in:
*A Complete Collection of Works*, Vol. 5 [in Russian], Fizmatgiz, Moscow (1937), pp. 72–80, 100–115.Google Scholar - 274.N. E. Zhukovskii, “On bird soaring,” in:
*A Complete Collection of Works*, Vol. 5 [in Russian], Fizmatgiz, Moscow (1937), pp. 49–59.Google Scholar