Journal of Mathematical Sciences

, Volume 210, Issue 3, pp 292–330 | Cite as

Some Classes of Integrable Problems in Spatial Dynamics of a Rigid Body in a Nonconservative Force Field

Article
  • 20 Downloads

Abstract

This paper is a review of some previous and new results on integrable cases in the dynamics of a three-dimensional rigid body in a nonconservative field of forces. These problems are stated in terms of dynamical systems with the so-called zero-mean variable dissipation. Finding a complete set of transcendental first integrals for systems with dissipation is a very interesting problem that has been studied in many publications. We introduce a new class of dynamical systems with a periodic coordinate. Since such systems possess some nontrivial groups of symmetries, it can be shown that they have variable dissipation whose mean value over the period of the periodic coordinate vanishes, although in various regions of the phase space there may be energy supply or scattering. The results obtained allow us to examine some dynamical systems associated with the motion of rigid bodies and find some cases in which the equations of motion can be integrated in terms of transcendental functions that can be expressed as finite combinations of elementary functions.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. A. Andronov, Collected Works [in Russian], Izd. AN SSSR, Moscow (1956).Google Scholar
  2. 2.
    A. A. Andronov and L. S. Pontryagin, “Rough systems,” Dokl. Akad. Nauk SSSR, 14, No. 5, 247–250 (1937).Google Scholar
  3. 3.
    A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Mayer, Qualitative Theory of Second-Order Dynamical Systems [in Russian], Nauka, Moscow (1966).Google Scholar
  4. 4.
    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
  5. 5.
    V. I. Arnold, “Hamiltonian nature of the Euler equations of the dynamics of a rigid body in an ideal fluid,” Usp. Mat. Nauk, 24, No. 3, 225–226 (1969).Google Scholar
  6. 6.
    V. I. Arnold, Mathematical Methods in Classical Mechanics [in Russian], Nauka, Moscow, (1989).CrossRefGoogle Scholar
  7. 7.
    V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], VINITI, Moscow (1985).Google Scholar
  8. 8.
    I. Bendikson, “On curves defined by differential equations,” Usp. Mat. Nauk, 9 (1941).Google Scholar
  9. 9.
    O. I. Bogoyavlenskii, “Some integrable cases of the Euler equations,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986).MathSciNetGoogle Scholar
  10. 10.
    A. D. Bryuno, The Local Method of Nonlinear Analysis of Differential Equations [in Russian], Nauka, Moscow (1979).Google Scholar
  11. 11.
    N. Bourbaki, Integration [Russian translation], Nauka, Moscow (1970).Google Scholar
  12. 12.
    D. V. Georgievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with a fixed point in R n,” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001).MathSciNetGoogle Scholar
  13. 13.
    D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations of a rigid body with a fixed point in R n,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002).MathSciNetGoogle Scholar
  14. 14.
    D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of a generalized gyroscope in R n,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 37–41 (2003).Google Scholar
  15. 15.
    D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,” J. Math. Sci., 154, No. 4, 449–461 (2008).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    V. V. Golubev, Lectures on the Analytical Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1950).Google Scholar
  17. 17.
    V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Heavy Solid Body with a Fixed Point [in Russian], Gostekhizdat, Moscow (1953).Google Scholar
  18. 18.
    D. N. Goryachev, “New integrable cases of the dynamical Euler equations,” Warsaw Univ. Izv., 3, 1–15 (1916).Google Scholar
  19. 19.
    D. M. Grobman, “Topological classification of neighborhoods of a singular point in n-dimensional space,” Mat. Sb., 56, No. 1, 77–94 (1962).MathSciNetGoogle Scholar
  20. 20.
    B. A. Dubrovin, B. A. Krichever, and S. P. Novikov, “Integrable Systems. I,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat., 4, 179–284 (1985).MathSciNetGoogle Scholar
  21. 21.
    V. V. Kozlov, Methods of Qualitative Analysis in Solid Dynamics [in Russian], Moscow Univ. Press, Moscow (1980).Google Scholar
  22. 22.
    V. V. Kozlov, “Integrability and nonintegrability in Hamiltonian mechanics,” Usp. Mat. Nauk, 38, No. 1, 3–67 (1983).Google Scholar
  23. 23.
    V. V. Kozlov and N. N. Kolesnikov, “On the integrability of Hamilton systems,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 88–91 (1979).Google Scholar
  24. 24.
    A. N. Kolmogorov, “General theory of dynamical systems and classical mechanics,” in: International Mathematical Congress in Amsterdam [in Russian], Fizmatgiz, Moscow (1961), pp. 187–208.Google Scholar
  25. 25.
    S. Levschetz, Differential Equations: Geometrical Theory [Russian translation], Izd. Inostr. Lit., Moscow (1961).Google Scholar
  26. 26.
    A. M. Lyapunov, “A new integrable case of the equations of motion of a solid body in fluid,” in: Collected Works [in Russian], Vol. I, Izd. AN SSSR (1954), pp. 320–324.Google Scholar
  27. 27.
    Yu. I. Manin, “Algebraic aspects of differential equations,” J. Sov. Math., 11, 1–128 (1979).CrossRefMATHGoogle Scholar
  28. 28.
    J. E. Marsden and M. McCracken, Cycle Birth Bifurcation and Its Applications [Russian translation], Mir, Moscow (1986).Google Scholar
  29. 29.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations [in Russian], Gostekhizdat, Moscow (1949).Google Scholar
  30. 30.
    Z. Nitecky, Introduction to Differential Dynamics [Russian translation], Mir, Moscow (1975).Google Scholar
  31. 31.
    S. P. Novikov and I. Shmeltzer, “Periodic solutions to Kirchoff equations for a free motion of a rigid body in fluid and Lusternik–Shnirelman–Morse extended theory,” Funct. Anal. Appl., 15, 54–66 (1981).Google Scholar
  32. 32.
    J. Palis and S. Smale, “Structural stability theorems,” in: Matematika: Sbornik Perevodov, 13, No. 2, 145–155 (1969).Google Scholar
  33. 33.
    J. Palis and W. De Melo, Geometric Theory of Dynamical Systems. An Introduction [Russian translation], Mir, Moscow (1986).Google Scholar
  34. 34.
    V. A. Pliss, “On roughness of differential equations on the torus,” Vestn. Leningr. Univ. Ser. Mat., 13, 15–23 (1960).MathSciNetGoogle Scholar
  35. 35.
    V. A. Pliss, Integral Sets of Periodic Systems of Differential Equations [in Russian], Nauka, Moscow (1967).Google Scholar
  36. 36.
    A. Poincar´e, On Curves Defined by Differential Equations, OGIZ, Moscow (1947).Google Scholar
  37. 37.
    A. Poincar´e, “New methods in celestial mechanics,” in: Selected Works, Vol. 1, 2, Nauka, Moscow (1971, 1972).Google Scholar
  38. 38.
    V. A. Samsonov and M. V. Shamolin, “On the problem of motion of a body in resistant media,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3, 51–54 (1989).Google Scholar
  39. 39.
    S. Smale, “Differentiable dynamical systems,” Usp. Mat. Nauk, 25, No. 1, 113–185 (1970).MathSciNetGoogle Scholar
  40. 40.
    V. A. Steklov, On the Motion of a Solid Body in Fluid [in Russian], Kharkiv (1893).Google Scholar
  41. 41.
    V. V. Trofimov, “Generalized Maslov classes of Lagrangian surfaces in symplectic manifolds,” Usp. Mat. Nauk, 43, No. 4, 169–170 (1988).Google Scholar
  42. 42.
    V. V. Trofimov and A. T. Fomenko, “Dynamical systems on linear representation orbits and complete integrability of some hydrodynamic systems,” Funkt. Anal. Pril., 17, No. 1, 31–39 (1983).MathSciNetGoogle Scholar
  43. 43.
    V. V. Trofimov and M. V. Shamolin, “Dissipative systems with nontrivial generalized Arnold–Maslov classes,” in: Abstracts of Lectures at the P. K. Rashevskii Seminar on Vector and Tensor Analysis, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 62 (2000).Google Scholar
  44. 44.
    Ph. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).Google Scholar
  45. 45.
    S. A.Chaplygin, Selected Works [in Russian], Nauka, Moscow (1976).Google Scholar
  46. 46.
    M. V. Shamolin, “Closed trajectories of different topological type in the problem of motion of a body in a resistant medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 52–56 (1992).Google Scholar
  47. 47.
    M. V. Shamolin, “A problem of motion of a body in a resistant medium,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 52–58 (1992).Google Scholar
  48. 48.
    M. V. Shamolin, “Classification of phase portraits in the problem of motion of a body in a resistant medium with linear damping moment,” Prikl. Mat. Mekh., 57, No. 4, 40–49 (1993).MathSciNetGoogle Scholar
  49. 49.
    M. V. Shamolin, “Methods of topographical Poincar´e systems and comparison systems: applications to specific systems of differential equations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 66–70 (1993).Google Scholar
  50. 50.
    M. V. Shamolin, “Dynamical systems on the plane: existence and uniqueness of trajectories with infinite points as limit sets,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 1, 68–71 (1993).Google Scholar
  51. 51.
    M. V. Shamolin, “A new two-parameter family of phase portraits in the problem of motion of a body in a medium,” Dokl. Ross. Akad. Nauk, 337, No. 5, 611–614 (1994).MathSciNetGoogle Scholar
  52. 52.
    M. V. Shamolin, “Definition of relative roughness and a two-parameter family of phase portraits in solid dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996).MathSciNetCrossRefGoogle Scholar
  53. 53.
    M. V. Shamolin, “Various types of phase portraits in the dynamics of rigid bodies interacting with resistant media,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996).MathSciNetGoogle Scholar
  54. 54.
    M. V. Shamolin, “Introduction to a problem of deceleration of a body in a resistant medium and a new two-parameter family of phase portraits,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 57–69 (1996).Google Scholar
  55. 55.
    M. V. Shamolin, “An integrable case in spatial dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997).Google Scholar
  56. 56.
    M. V. Shamolin, “Spatial topographic Poincar´e systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997).MathSciNetCrossRefGoogle Scholar
  57. 57.
    M. V. Shamolin, “On integrability in terms of transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998).MathSciNetCrossRefGoogle Scholar
  58. 58.
    M. V. Shamolin, “A family of portraits with limit cycles in the dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998).Google Scholar
  59. 59.
    M. V. Shamolin, “Some classes of particular solutions in the dynamics of rigid bodies interacting with media,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178–189 (1999).Google Scholar
  60. 60.
    M. V. Shamolin, “New Jacobi integrable cases in the dynamics of a rigid body interacting with media,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999).Google Scholar
  61. 61.
    M. V. Shamolin, “On roughness of dissipative systems and relative roughness and nonroughness of systems with variable dissipation,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999).MathSciNetCrossRefGoogle Scholar
  62. 62.
    M. V. Shamolin, A new family of phase portraits in spatial dynamics of a rigid body interacting with media,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000).MathSciNetGoogle Scholar
  63. 63.
    M. V. Shamolin, “On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000).MathSciNetCrossRefGoogle Scholar
  64. 64.
    M. V. Shamolin, “Jacobi integrability in the problem of a four-dimensional rigid body in a resistant medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000).Google Scholar
  65. 65.
    M. V. Shamolin, “Integrable cases in spatial dynamics of rigid bodies,” Prikl. Mekh., 37, No. 6, 74–82 (2001).MathSciNetMATHGoogle Scholar
  66. 66.
    M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in an incoming flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001).Google Scholar
  67. 67.
    M. V. Shamolin, “On the integrability of some classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002).MathSciNetCrossRefGoogle Scholar
  68. 68.
    M. V. Shamolin, “Geometrical representation of motion in a problem of interaction of a body with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004).MathSciNetMATHGoogle Scholar
  69. 69.
    M. V. Shamolin, “A completely integrable case in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005).MathSciNetGoogle Scholar
  70. 70.
    M. V. Shamolin, “Comparison of Jacobi integrable cases in plane and spatial dynamics of jet flow past a body,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005).MathSciNetMATHGoogle Scholar
  71. 71.
    M. V. Shamolin, “An integrable case of dynamical equations on so(4) × R n,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005).MathSciNetCrossRefGoogle Scholar
  72. 72.
    M. V. Shamolin, “Complete integrability of the equations of motion of a spatial pendulum in a medium flow with rotational derivatives of the torque produced by the medium taken into account,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 3, 187–192 (2007).Google Scholar
  73. 73.
    M. V. Shamolin, “A case of total integrability in the dynamics on the tangent bundle of the twodimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007).MathSciNetCrossRefGoogle Scholar
  74. 74.
    M. V. Shamolin, Methods of Analysis of Dynamical Systems with Variable Dissipation in the Dynamics of Solids [in Russian], Ekzamen, Moscow (2007).Google Scholar
  75. 75.
    M. V. Shamolin, Some Problems of Differential and Topological Diagnostics [in Russian], Ekzamen, Moscow (2007).Google Scholar
  76. 76.
    M. V. Shamolin, “A three-parameter family of phase portraits in the dynamics of rigid bodies interacting with media,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008).MathSciNetGoogle Scholar
  77. 77.
    D. Arrowsmith and K. Place, Ordinary Differential Equations: Qualitative Theory and Applications [Russian translation], Mir, Moscow (1986).Google Scholar
  78. 78.
    M. V. Shamolin, “Some classical problems in three dimensional dynamics of a rigid body interacting with a medium,” in: Proc. ICTACEM’98, Kharagpur, India, Dec. 1–5, 1998, CD, Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998).Google Scholar
  79. 79.
    M. V. Shamolin, “Structural stability in 3D dynamics of a rigid,” in: Proc. WCSMO-3, Buffalo, NY, May 17–21, 1999, CD, Buffalo, NY (1999).Google Scholar
  80. 80.
    M. V. Shamolin, “New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: Proc. 16th IMACS World Cong. 2000, Lausanne, Switzerland, August 21–25, CD, EPFL, Lausanne (2000).Google Scholar
  81. 81.
    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).MathSciNetCrossRefGoogle Scholar
  82. 82.
    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).MathSciNetCrossRefMATHGoogle Scholar
  83. 83.
    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).MathSciNetCrossRefMATHGoogle Scholar
  84. 84.
    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, Dec. 12–15, 2005, Vol. 1, Tech. Univ. Lodz (2005), pp. 429–436.Google Scholar
  85. 85.
    M. V. Shamolin, “The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Proc. 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007 ), Lodz, Poland, Dec. 17–20, 2007, Vol. 1, Tech. Univ. Lodz (2007), pp. 415–422.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

Personalised recommendations