Skip to main content
Log in

New Integrable Cases and Families of Portraits in the Plane and Spatial Dynamics of a Rigid Body Interacting with a Medium

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

REFERENCES

  1. A. A. Andronov, Collection of Works [in Russian ], Moscow, Izd. Akad. Nauk SSSR (1956).

    Google Scholar 

  2. A. A. Andronov and E. A. Leontovich, “Certain cases of the de endency of limit cycles on the parameter, ”Uchenye Zapiski Gork. Gos. Univ., No. 6 (1937).

  3. A. A. Andronov and E. A. Leontovich, “On the theory of changes in the qualitative structure of the partition of a lane into trajectories, ”Dokl. Akad. Nauk SSSR, 21, No. 9 (1938).

    Google Scholar 

  4. A. A. Andronov and E. A. Leontovich, “Bifurcation of limit cycles from a structurally unstable focus or center and from a structurally unstable limit cycle, ”Ma.Sb., 40, No. 2 (1956).

  5. A. A. Andronov and E. A. Leontovich, “On the bifurcation of limit cycles from a separatrix loo and from the separatrix of the equilibrium state of the saddle-node ty e, ”Ma. Sb., 48, No. 3 (1959).

    Google Scholar 

  6. A. A. Andronov and E. A. Leontovich, “Dynamical systems of the rst degree of structural instability on the lane, ”Ma. Sb., 68, No. 3, 328–372 (1965).

    Google Scholar 

  7. A. A. Andronov, and E. A. Leontovich, “Suffcient conditions for the first-degree structural insta-bility of a dynamical system on the lane, ”Differents. Urav., 6, No. 12, 2121–2134 (1970).

    Google Scholar 

  8. A. A. Andronov and L. S. Pontryagin, “Rough systems, ”Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937).

    Google Scholar 

  9. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillations [in Russian ], Nauka, Moscow (1981).

    Google Scholar 

  10. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, The Qualitative Theory of Second-Order Dynamical Systems [in Russian ], Nauka, Moscow (1966).

    Google Scholar 

  11. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Theory of Bifurcations of Dynamical Systems on the Plane [in Russian ], Nauka, Moscow (1967).

    Google Scholar 

  12. D. V. Anosov, “Geodesic. flows on closed Riemannian manifolds of negative curvature, ”Tr. Mat. Inst. Akad. Nauk SSSR, 90 (1967).

  13. P. Appel, Theoretical Mechanics (in two volumes) [in Russian ], Fizmatgiz, Moscow (1960).

    Google Scholar 

  14. S. Kh. Aranson, “Dynamical systems on two-dimensional manifolds, ”In: Proc. 5th Intern. Conf. on Nonlinear Vibrations. Vol 2 [Russian translation ], Inst. Mat. Akad. Nauk Ukr. SSSR, Kiev (1970),.46–52.

    Google Scholar 

  15. S. Kh. Aranson and V. Z. Grines, “Topological classi cation of. ows on closed two-dimensional manifolds, ”Usp. Ma. Nauk, 41, No. 1, 149–169 (1986).

    Google Scholar 

  16. V. I. Arnold, “The Euler equations of dynamics of rigid bodies in an ideal. uid are Hamiltonian, ”Usp. Ma. Nauk, 24, No. 3, 225–226 (1969).

    Google Scholar 

  17. V. I. Arnold, Supplementary Chapters of he Theory of Ordinary Di. erential Equations [in Russian ], Nauka, Moscow (1978).

  18. V. I. Arnold, Ordinary Di. erential Equations [in Russian ], Nauka, Moscow (1984).

  19. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian ], Nauka, Moscow (1989).

  20. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, “Mathematical as ects of classical and celestial mechanics, ”In: Progress in Science and Technology, Series on Contemporary problems in Mathemattics, Fundamental Directions, Dynamical Systems–3, All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1985).

    Google Scholar 

  21. Yu. A. Arkhangel 'skii, Analytical Dynamics of Rigid Bodies [in Russian ], Nauka, Moscow (1977).

    Google Scholar 

  22. N. N. Bautin, “On the number of limit cycles that bifurcate from the equilibrium state of the focus or center type under a change of coeffcients, ”Mat. Sb., 30(72), No. 1 (1952).

    Google Scholar 

  23. N. N. Bautin, “On the approximation and structural stability of the space of arameters of a dynamical system, ”In: Proc. 5th Intern. Conf. on Nonlinear Vibrations [in Russian ], Kiev (1970), 75–82.

  24. N. N. Bautin, “Certain methods for the qualitative study of dynamical systems involving the rotation of a field, ”Prikl. Mat. Mekh., 37, No. 6, 984–989 (1973).

    Google Scholar 

  25. N. N. Bautin and E. A. Leontovich, Methods and Techniques for he Qualita ive Study of Dynamical Systems on the Plane [in Russian ] Nauka, Moscow (1976).

    Google Scholar 

  26. V. V. Beletskii, The Motion of an Artificial Satellite Relative o he Center of Mass [in Russian ], Nauka, Moscow (1965).

    Google Scholar 

  27. V. V. Beletskii, The Motion of a Satellite Relative o he Center of Mass in the Gravitational Field [in Russian ], Izd. Mosk. Gos. Univ., Moscow (1975).

    Google Scholar 

  28. V. V. Beletskii and A. M. Yanshin, The Effect of Aerodynamic Forces on he Rotational Motion of Artificial Satellites [in Russian ], Naukova Dumka, Kiev (1984).

    Google Scholar 

  29. G. Birkho., Dynamical Systems [Russian translation ], Gostekhizdat, Moscow –Leningrad (1941).

    Google Scholar 

  30. R. L. Bishop, Oscillations [Russian translation ], Nauka, Moscow (1986).

    Google Scholar 

  31. I. T. Borisenok, B. Ya. Lokshin, and V. A. Privalov, “On the dynamics of atmospheric. flight of axially symmetric rotating bodies, ”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 35–42 (1984).

    Google Scholar 

  32. A. D. Bruno, A Local Method for he Nonlinear Analysis of Di. erential Equations [in Russian ], Nauka, Moscow (1979).

  33. N. Bourbaki, Integration [Russian translation ], Nauka, Moscow (1970).

    Google Scholar 

  34. N. Bourbaki, Lie Groups and Algebras [Russian translation ], Mir, Moscow (1972).

    Google Scholar 

  35. N. N. Bukhgol' ts, A Principal Course in Theoretical Mechanics, Vols. 1, 2 [in Russian ], Nauka, Moscow (1972).

    Google Scholar 

  36. G. S. Byushgens and R. V. Studnev, The Dynamics of Longitudinal and Lateral Motions [in Russian ], Mashinostroenie, Moscow (1969).

    Google Scholar 

  37. G. S. Byushgens and R. V. Studnev, The Dynamics of Three-Dimensional Motion of an aircraft [in Russian ], Mashinostroenie (1988).

  38. F. R. Gantmakher, Lec ures on Analytical Mechanics [in Russian ], Nauka, Moscow (1960).

  39. V. V. Golubev, Lectures on the Analytical Theory of Di. erential Equations [in Russian ], Gostekhiz-dat, Moscow –Leningrad (1950).

  40. V. V. Golubev, Lectures on the Integration of Equations of Motion of a Massive Rigid Body in the Neighborhood of a Stationary Point [in Russian ], Gostekhizdat, Moscow –Leningrad (1953).

    Google Scholar 

  41. G. V. Gorr, L. V. Kudryashova, and L. A. Stepanova, Classical Problems of the Dynamics of Rigid Bodies [in Russian ], Naukova Dumka, Kiev (1978).

    Google Scholar 

  42. D. N. Goryachev, “Newcases of integrability of dynamic Euler equations, ”Izv. Varshavsk. Univ. Book 3, 1–15 (1916).

    Google Scholar 

  43. I. S. Gradshtein and I. M. Ryzhik Tables of Integrals, Sums of Series and Derivatives [in Russian ], Gostekhizdat, Moscow (1963).

    Google Scholar 

  44. D. M. Grobman, “On the homeomor hism of systems of di. erential equations, ”Dokl. Akad. Nauk SSSR, 128, No. 5, 880–881 (1962).

    Google Scholar 

  45. D. M. Grobman, “Topological classi cation of neighborhoods of a singular oint in the n-dimensional s ace, ”Ma. Sb., 56, No. 1, 77–94 (1962).

    Google Scholar 

  46. M. I. Gurevich, The Theory of Jets of an Ideal Fluid [in Russian ], Nauka, Moscow (1979).

    Google Scholar 

  47. G. Dyulak, On Limit Cycles [in Russian ], Nauka, Moscow (1980).

    Google Scholar 

  48. V. A. Eroshin, “The immersion of a disk in a compressible. uid at an angle to the free surface, ”Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 142–144 (1983).

    Google Scholar 

  49. V. A. Eroshin, The Pene ration of an Elastic Cylinder into Water at High Speed [in Russian ], Preprint No. 5, Institute of Mechanics of Moscow State University (1991).

  50. V. A. Eroshin, “Ex erimental study of the entry of an elastic cylinder into water at a high s eed, ”Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 20–30 (1992).

    Google Scholar 

  51. V. A. Eroshin, V. A. Privalov, and V. A. Samsonov, “Two model roblems on the motion of a body in a resisting medium, ”In: Collection of Research and Me hodological Papers on Theoretical Mechanics [in Russian ], No. 18, Nauka, Moscow (1987),. 75–78.

    Google Scholar 

  52. V. A. Eroshin, V. A. Samsonov, and M. V. Shamolin, “A model roblem on the deceleration of a body in a resisting medium under a jet. ow past this body, ”Izv. Ross. Akad. Nauk. Mekh. Zhidk. Gaza, No. 3, 23–27 (1995)

    Google Scholar 

  53. N. E. Zhukovskii, “On the fall of light, oblong bodies rotating about their longitudinal axis, ” in Complete Collection of Works [in Russian ], Vol. 5, Fizmatgiz, Moscow (1937),. 72–80, 100–115.

    Google Scholar 

  54. N. E. Zhukovskii, “On bird hovering, ”In: Complete Collection of Works [in Russian ], Vol. 5, Fizmatgiz, Moscow (1937),. 49–59.

    Google Scholar 

  55. V. F. Zhuravlev and D. M. Klimov, Applied Methods in the Theory of Oscillations, Nauka, Moscow (1988).

    Google Scholar 

  56. Yu. F. Zhuravlev, “The immersion of a disk into a liquid at an angle to the free surface, ”In: Collection of Works on Hydrodynamics [in Russian ], Central Aerohydrodynamic Institute, Moscow (1959),. 164–167.

    Google Scholar 

  57. G. Seifert and V. Trelfall, Topol ogy [Russian translation ], Gostekhizdat, Moscow –Leningrad (1938).

    Google Scholar 

  58. A. Yu. Ishlinskii, Orientation, Gyroscopes, and Inertial Navigation [in Russian ], Nauka, Moscow (1976).

    Google Scholar 

  59. A. B. Katok, “Dynamical systems with hy erbolic structures, ”In: The 9th Summer Mathematical School [in Russian ], Kiev (1972),. 125–211.

  60. V. V. Kozlov, Methods of Qualitative Analysis of he Dynamics of Rigid Bodies [in Russian ], MGU, Moscow (1980).

    Google Scholar 

  61. V. V. Kozlov, “Hydrodynamics of Hamiltonian systems, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 6, 10–22 (1983).

    Google Scholar 

  62. V. V. Kozlov, “Remarks on stationary vortex motion of a continuous medium, ”Prikl. Mat. Mekh., 47, No. 2, 341–342 (1983).

    Google Scholar 

  63. V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics, ”Usp. Ma. Nauk, 38, No. 1, 3–67 (1983).

    Google Scholar 

  64. V. V. Kozlov, “On the roblem of rotation of a rigid body in a magnetic eld, ”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 6, 28–33 (1985).

  65. V. V. Kozlov, “On the fall of a massive rigid body in an ideal. uid, ”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 5, 10–17 (1989).

    Google Scholar 

  66. V. V. Kozlov, “A vortex theory of a gyroscope, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 4, 56–62 (1990).

    Google Scholar 

  67. V. V. Kozlov, “On the roblem of the fall of a massive rigid body in a resisting medium, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 1, 79–87 (1990).

    Google Scholar 

  68. V. V. Kozlov and D. A. Onishchenko, “Nonintegrability of Kircho. equations, ”Dokl. Akad. Nauk SSSR, 266, No. 6, 1298–1300 (1982).

    Google Scholar 

  69. N. N. Kolesnikov, “Natural systems with solvable grou of symmetries, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 5, 99–103 (1978).

    Google Scholar 

  70. A. N. Kolmogorov, “The general theory of dynamical systems and classical mechanics, ”In: The International Mathematical Congress in Amsterdam [in Russian ], Fizmatgiz, Moscow (1961),. 187–208.

    Google Scholar 

  71. M. A. Krasnosel 'skii, A. I. Perov, A. I. Povolotskii, and P. P. Zabreiko,Vec or Fields on he Plane [in Russian ], Moscow, Fizmatgiz (1963).

    Google Scholar 

  72. N. M. Krylov and N. N. Bogolyubov, New Me hods of Nonlinear Mechanics [in Russian ], ONTI, Moscow –Leningrad (1934).

    Google Scholar 

  73. N. N. Krylov and N. N. Bogolyubov, An Introduc ion o Nonlinear Mechanics [in Russian ], Akad. Nauk SSSR, Moscow (1937).

    Google Scholar 

  74. G. Lamb, Hydrodynamics [Russian translation ], Fizmatgiz, Moscow (1947).

    Google Scholar 

  75. L. D. Landau and E. M. Lifshitz, Mechanics [in Russian ], Nauka, Moscow (1968).

    Google Scholar 

  76. S. Lefchetz, Geometric Theory of Differential Equations [Russian translation ], Inostrannaya Literatura, Moscow (1961).

    Google Scholar 

  77. E. A. Leontovich, “On the definition of a rough dynamical system, ”In: Nonlinear Vibrations Problems, Second Conference on Nonlinear Vibrations, Warsaw (1964).

  78. E. A. Leontovich and A. G. Maier, “On trajectories determining the qualitative structure of the artition of a sphere into trajectories, ”Dokl. Akad. Nauk SSSR, 14, No. 5 (1937).

    Google Scholar 

  79. E. A. Leontovich and A. G. Maier, “On the attern determining the topological structure of the artition into trajectories, ”Dokl. Akad. Nauk SSSR, 103, No. 4, 557–560 (1955).

    Google Scholar 

  80. E. A. Leontovich and L. P. Shil' nikov. Theory of Bifurca ions of Dynamical Systems: the State-of-Art. Qualitative Methods of the Theory of Nonlinear Vibrations [in Russian ], Vol. 2, Inst. Mat. AN Ukr. SSR, Kiev (1970),. 282–291.

    Google Scholar 

  81. G. W. Lych, Classical Mechanics [Russian translation ], Inostrannaya Literatura, Moscow (1961).

    Google Scholar 

  82. B. Ya. Lokshin, “On one kind of motion of a fast-rotating body in the air, ”Ves n. MGU, Ser 1, Mat., Mekh., No. 6, 93–98 (1970).

    Google Scholar 

  83. B. Ya. Lokshin, “On the stability of lane motion of a fast-rotating symmetric body in the atmosphere, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 4, 113–118 (1971).

    Google Scholar 

  84. B. Ya. Lokshin, “On the helicoidal motion of a fast-rotating rigid symmetric body in the air, ”Ves n. MGU, Ser. 1, Ma., Mekh., No. 4, 79–86 (1973).

    Google Scholar 

  85. B. Ya. Lokshin, “On the stability of stationary motions of a fast-rotating symmetric body in the air,”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 18–24 (1976).

    Google Scholar 

  86. B. Ya. Lokshin and O. Yu. Cherkasov, “On the structure of optimal trajectories of a rotating rigid body in a resisting medium,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 1, 63–68 (1990).

    Google Scholar 

  87. B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, An Introduction to the Problem of Motion of a Body in a Resisting Medium [in Russian ], MGU, Moscow (1986).

  88. B. Ya. Lokshin, V. A. Privalov, and V. A. Samsonov, An Introduc ion o he Problem of Motion of a Material Point and a Body in a Resisting Medium [in Russian ], MGU, Moscow (1992).

    Google Scholar 

  89. B. Ya. Lokshin, Yu. M. Okunev, V. A. Samsonov, and M. V. Shamolin,“Certain integrable cases of three-dimensional vibrations of a rigid body in a resisting medium,”In: Abstracts of Reports of 21st Readings in Astronautics(Moscow, January 28–31, 1997)[in Russian ], Inst. Istorii Estestvoznaniya Tekhniki Ross. Akad. Nauk (IIET RAN), Moscow (1997),. 82–83.

    Google Scholar 

  90. A. M. Lyapunov, “A newcase of integrability of equations of motion of a rigid body in a fluid,”In: Complete Collection of Works [in Russian ], Vol. 1, Akad. Nauk SSSR, Moscow (1954),. 320–324.

    Google Scholar 

  91. I. G. Malkin, Certain Problems of he Theory of Nonlinear Vibrations [in Russian ], Gostekhteorizdat, Moscow (1956).

    Google Scholar 

  92. Yu. I. Manin, “Algebraic aspects of nonlinear differential equations,”In: Progress in Science and Technology, Series on Contemporary Problems in Mathematics [in Russian ], All Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1978),. 5–112.

    Google Scholar 

  93. A. P. Markeev, “On the integrability of the roblem of the rolling motion of a ball with a multicon-nected cavity filled with an ideal fluid,”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 1, 64–65 (1986).

    Google Scholar 

  94. A. P. Markeev, Theoretical Mechanics [in Russian ], Nauka, Moscow (1990).

    Google Scholar 

  95. J. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications [Russian translation ], Mir, Moscow (1986).

    Google Scholar 

  96. W. Miller, Symmetry and Separation of Variables [Russian translation ], Mir, Moscow (1981).

    Google Scholar 

  97. N. N. Moiseev, A symptotic Methods of Nonlinear Mechanics [in Russian ], Nauka, Moscow (1969).

    Google Scholar 

  98. N. N. Moiseev and V. V. Rumyantsev, Dynamics of Bodies with Cavities Filled with a Fluid [in Russian ], Nauka, Moscow (1965).

    Google Scholar 

  99. Yu. I. Neimark, “The structure of motions of a dynamical system in a neighborhood of a homoclinic curve,”In: 5th Summer Mathematical School [in Russian ], Kiev (1968),. 400–435.

  100. V. V. Nemytskii and V. V. Stepanov, The Qualitative Theory of Differential Equations [in Russian ], Gostekhizdat, Moscow –Leningrad (1949).

  101. Z. Nitecki, Introduction o Differential Dynamics [Russian translation ], Mir, Moscow (1975).

    Google Scholar 

  102. S. P. Novikov and I. Shmel' tser, “Periodic solutions to the Kircho. equations of free motion of a rigid body and the ideal fluid and the exended Lusternik–Shnirel' man–Morse (LSM)theory. I,”Funkts. Anal. Pril., 15, No. 3, 54–66 (1981).

    Google Scholar 

  103. Yu. M. Okunev and V. A. Sadovnichii, “Model dynamical systems of one of the roblems of external ballistics and their analytical solutions,”In: Problems of Modern Mechanics [in Russian ], Izd. Mosk. Gos. Univ., Moscow (1998),. 28–46.

  104. Yu. M. Okunev, V. A. Privalov, and V. A. Samsonov, “Certain roblems of motion of a body in a resisting medium,”In: Proc. All-Union Conf. on Nonlinear Phenomena [in Russian ], Nauka, Moscow (1991),. 140–144.

    Google Scholar 

  105. Yu. M. Okunev, V. A. Sadovnichii, V. A. Samsonov, and G. G. Chernyi, “A complex for modeling flight dynamics problems,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 6, 66–75 (1996).

    Google Scholar 

  106. J. Palais and S. Smale, “Theorems on structural stability,”In: Sb. Per. Ma., 13, No. 2, 145–155 (1969).

    Google Scholar 

  107. J. Palis and W. De Melo, Geometric Thory of Dynamical Systems: An Introduction [Russian trans-lation ], Mir, Moscow (1986).

    Google Scholar 

  108. A. M. Perelomov, “Several remarks on integration of equations of motion of a rigid body in an ideal fluid,”Funkts. Anal. Prilozh., 15, No. 2, 83–85 (1981).

    Google Scholar 

  109. I. G. Petrovskii, Lectures in the Theory of Ordinary Differential Equations [in Russian ], Gostekhiz-dat, Moscow –Leningrad (1952).

    Google Scholar 

  110. V. A. Pliss, “On the roughness of differential equations assigned on a torus,”Vestn. LGU, Ser. Ma., 13, 15–23 (1960).

    Google Scholar 

  111. V. A. Pliss, Nonlocal Problems of the Theory of Vibrations [in Russian ], Nauka, Moscow –Leningrad (1964).

    Google Scholar 

  112. V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian ], Nauka, Moscow (1967).

    Google Scholar 

  113. V. A. Pliss, “On the stability of an arbitrary system with respect to erturbations that are small in the sense of Smale,”Di. erents. Urav. 16, No. 10, 1891–1892 (1980).

  114. V. A. Privalov and V. A. Samsonov, “On the stability of motion of an autorotating body in the flow of a medium,”Izv. Akad. Nauk SSSR, Mekh. Tverd. Tela, No. 2, 32–38 (1990).

    Google Scholar 

  115. H. Poincar ´e, On Curves Defined by Differential Equations [Russian translation ], OGIZ, Moscow –Leningrad (1947).

    Google Scholar 

  116. H. Poincar ´e, “Newmethods in celestial mechanics,”in: H. Poincar ´e Selected Works [Russian translation ], Vols. 1, 2, Nauka, Moscow (1971, 1972).

    Google Scholar 

  117. H. Poincar ´e, On Science [Russian translation ], Nauka, Moscow (1983).

    Google Scholar 

  118. R. Reissing, G. Sansonet, and R. Contie, The Qualitative Theory of Nonlinear Differential Equations [Russian translation ], Nauka, Moscow (1974).

    Google Scholar 

  119. S. T. Sadetov, “Conditions for the integrability of Kircho. equations,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 3, 56–62 (1990).

    Google Scholar 

  120. T. V. Sal' nikov, “On the integrability of Kirchoff equations in the symmetric case,”Ves n. MGU, Ser. 1, Mat., Mekh., No. 4, 68–71 (1985).

    Google Scholar 

  121. V. A. Samsonov “On the stability of solutions to systems of differential equations in certain cases,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 5, 74–78 (1962).

    Google Scholar 

  122. V. A. Samsonov, “On quasi-stationary motions of mechanical systems,”Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 1, 32–5 (1978).

    Google Scholar 

  123. V. A. Samsonov and M. V. Shamolin, “On the problem of motion of a body in a resisting medium,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 3, 51–54, 105 (1989).

    Google Scholar 

  124. V. A. Samsonov and M. V. Shamolin, “A model roblem of motion of a body in a medium under a jet flow past this body,”In: Research Report of the Institute of Mechanics of Moscow State University, No. 3969 [in Russian ], Moscow (1990).

  125. V. A. Samsonov and M. V. Shamolin, “On the roblem of the deceleration of a body in a medium under a jet flow past this body,”In: Research Report of the Institute of Mechanics of Moscow State University, No. 4141 [in Russian ], Moscow (1991).

  126. V. A. Samsonov, V. A. Eroshin, G. A. Konstantinov, and V. M. Makarshin, “Two model problems of motion of a body in a medium under a jet flow past this body,”In: Research Repor of he Institute of Mechanics of Moscow State University, No. 3427 [in Russian ], Moscow (1987).

  127. V. A. Samsonov, M. V. Shamolin, V. A. Eroshin, and V. M. Makarshin, “Mathematical modeling in the problem of the deceleration of a body in a resisting medium under a jet. owpast this body,”In: Research Report of the Institute of Mechanics of Moscow State University, No. 4396 [in Russian ], Moscow (1995).

  128. G. Sansonet, Ordinary Differential Equations [Russian translation ], Inostrannaya Literatura, Moscow (1954).

    Google Scholar 

  129. L. I. Sedov, Continuum Mechanics [in Russian ]Vol. 1, Nauka, Moscow (1983); Vol. 2, Nauka, Moscow (1984).

    Google Scholar 

  130. J. L. Synge, Classical Dynamics [Russian translation ], Fizmatgiz, Moscow (1963).

    Google Scholar 

  131. S. Smale, “Rough systems are not dense,”Sb. Per. Mat., 11, No. 4, 107–112 (1967).

    Google Scholar 

  132. S. Smale, “Differentiable dynamical systems,”Usp. Ma. Nauk, 25, No. 1, 113–185 (1970).

    Google Scholar 

  133. V. M. Starzhinskii, Applied Methods of Nonlinear Vibrations [in Russian ], Nauka (1977).

  134. V. A. Steklov, On the Motion of a Rigid Body in a Fluid [in Russian ], Khar' kov (1893).

  135. V. V. Stepanov, A Course in Differential Equations [in Russian ], Fizmatgiz, Moscow (1959).

    Google Scholar 

  136. G. K. Suslov, Theoretical Mechanics [in Russian ], Gostekhizdat, Moscow (1946).

    Google Scholar 

  137. V. V. Sychev, A. I. Ruban, V. V. Sychev, and G. L. Korolev, Asymptotic Theory of Detached Flows [in Russian ], Nauka, Moscow (1987).

    Google Scholar 

  138. V. G. Tabachnikov, “Stationary characteristics of wings at low speeds over the whole range of angles of attack,”In: Trudy Centr. Aerohydrodyn. Inst., No. 1621, Moscow (1974),. 18–24.

    Google Scholar 

  139. Ya. V. Tatarinov, Lectures on Classical Dynamics [in Russian ], Izd. Mosk. Gos. Univ., Moscow (1984).

    Google Scholar 

  140. V. V. Tro mov, “Euler 's equations on nite-dimensional solvable Lie groups,”Izv. Akad. Nauk SSSR. Ser. Mat., 44, No. 5, 1191–1199 (1980).

    Google Scholar 

  141. E. T. Whittaker, Analytical Dynamics [Russian translation ], ONTI, Moscow (1937).

    Google Scholar 

  142. P. Hartman, Ordinary Differential Equations [Russian translation ], Mir, Moscow (1970).

    Google Scholar 

  143. S. A. Chaplygin, “On the motion of massive bodies in an incompressible fluid,”In: Complete Collection of Works [in Russian ], Vol. 1, Izd. Akad. Nauk SSSR, Leningrad (1933),. 133–135.

    Google Scholar 

  144. S. A. Chaplygin, Selected Works [in Russian ], Nauka, Moscow (1976).

    Google Scholar 

  145. 145. F. L. Chernous' ko, L. D. Akulenko, and B. N. Sokolov, Control of Vibrations [in Russian ], Nauka, Moscow (1980).

    Google Scholar 

  146. M. V. Shamolin, “Closed trajectories of distinct topological ty es in the problem of motion of a body in a resisting medium,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 2, 52–56, 112 (1992).

    Google Scholar 

  147. M. V. Shamolin, “On the problem of motion of a body in a resisting medium,”Ves n. MGU, Ser. 1, Mat., Mekh., No. 1, 52–58, 112 (1992).

    Google Scholar 

  148. M. V. Shamolin, “Classification of hase portraits in the problem of motion of a body in a resisting medium under the existence of a linear damping moment,”Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993).

    Google Scholar 

  149. M. V. Shamolin, “Application of Poincar ´e ma systems and reference systems in some particular systems of differential equations,”Ves n. MGU, Ser.1, Ma., Mekh., No. 2, 66–70, 113 (1993).

    Google Scholar 

  150. M. V. Shamolin, “Existence and uniqueness of trajectories having infinitely remote oints as limit sets for dynamical systems on the plane,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 1, 68–71, 112 (1993).

    Google Scholar 

  151. M. V. Shamolin, “Newtwo-parameter family of hase ortraits in the roblem of motion of a body in a medium,”Dokl. Ross. Akad. Nauk, 337, No. 5, 611–614 (1994).

    Google Scholar 

  152. M. V. Shamolin, “The definition of relative structural stability and a two-parameter family of hase portraits in the dynamics of a rigid body,”Usp. Ma. Nauk, 51, No. 1, 175–176 (1996).

    Google Scholar 

  153. M. V. Shamolin, “Periodic and Poisson-stable trajectories in the roblem of motion of a body in a resisting medium,”Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55–63 (1996).

    Google Scholar 

  154. M. V. Shamolin, “A variety of types of hase ortraits in the dynamics of a rigid body interacting with a resisting medium,”Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996).

    Google Scholar 

  155. M. V. Shamolin, “An introduction to the roblem on the deceleration of a body in a resisting medium and a new two-parametric family of phase portraits,”Ves n. MGU, Ser. 1, Ma., Mekh., No. 4, 57–69 (1996).

    Google Scholar 

  156. M. V. Shamolin, “On the integrable case in the three-dimensional dynamics of a rigid body inter-acting with a medium,”Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997).

    Google Scholar 

  157. M. V. Shamolin, “S atial Poincar ´e ma systems and reference systems,”Usp. Ma. Nauk, 52, No. 3, 177–178 (1997).

    Google Scholar 

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

    Google Scholar 

  159. M. V. Shamolin, “A family of ortraits with limit cycles in the lane dynamics of a rigid body interacting with a medium,”Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  162. M. V. Shamolin and S. V. Tsy tsin, “Analytical and numerical study of trajectories of motion of a body in a resisting medium,”In: Research Report of the Institute of Mechanics of Moscow State University No. 4289 [in Russian ], Moscow (1993).

  163. D. Arrowsmith and C. Place, Ordinary Differential Equations. A Qualitative Approach with Applications [Russian translation ], Mir, Moscow (1986).

    Google Scholar 

  164. C. Jacobi, Lectures on Dynamics [in Russian ], ONTI, Moscow –Leningrad (1936).

  165. M. V. Jacobson, “On smooth ma ings of a circle into itself,”Ma. Sb., No. 85, 183–188 (1975).

    Google Scholar 

  166. A. Yu. Ishlinsky and D. M. Klimov, “Some as ects of the solution of the main roblem of inertial navigation,”J. Inst. Navig., 23, No. 4 (1970)

  167. M. Peixoto, “On structural stability,”Ann. Math., (2), 69, 199–222 (1959).

    Google Scholar 

  168. M. Peixoto, “Structural stability on two-dimensional manifolds,”Topol ogy, 1, No. 2, 101–120 (1962).

    Google Scholar 

  169. M. Peixoto, “On an a roximation theorem of Ku ka and Smale,”J. Diff. Eq., 3214–227 (1966).

    Google Scholar 

  170. L. Prandtl and A. Betz, Ergebnisse der Aerodinamishen Versuchsastal zu G ¨ottingen, b. 4 Liefrung. M ¨unchen–Berlin; R. Oldenbourg (1932).

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

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shamolin, M.V. New Integrable Cases and Families of Portraits in the Plane and Spatial Dynamics of a Rigid Body Interacting with a Medium. Journal of Mathematical Sciences 114, 919–975 (2003). https://doi.org/10.1023/A:1021865626829

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021865626829

Keywords

Navigation