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New Cases of Integrability of Equations of Motion of a Rigid Body in the n-Dimensional Space

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References

  1. V. I. Arnold, V. V. Kozlov, and A. I. Nejshtadt, Mathematical Aspects of Classical and Celestial Mechanics [in Russian], Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya, 3, 5–304 (1985).

  2. A. V. Belyaev, “On the motion of a multi-dimensional body with a clamped point in the gravity force field,” Mat. Sb., 114, No. 3, 465–470 (1981).

    MathSciNet  Google Scholar 

  3. O. I. Bogoyavlenskii, “Some integrable cases of Euler equation,” Dokl. Akad. Nauk SSSR, 287, No. 5, 1105–1108 (1986).

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. N. Bourbaki, Lie Groups and Lie Algebras [Russian translation], Nauka, Moscow (1970).

    MATH  Google Scholar 

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

    Google Scholar 

  7. D. V. Georgievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with a fixed point in ℝn,” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001).

    MathSciNet  Google Scholar 

  8. D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with a fixed point in ℝn,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002).

    Google Scholar 

  9. D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion for a generalized gyroscope in ℝn,” Vestn. MGU, Ser. 1, Mat., Mekh., 5, 37–41 (2003).

  10. D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,” J. Math. Sci., 154, No. 4, 449–461 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  11. D. V. Georgievskii and M. V. Shamolin, “Sessions of the Workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, “Topical Problems of Geometry and Mechanics” Named after V. V. Trofimov,” J. Math. Sci., 165, No. 6, 607–615 (2010).

    Article  MATH  Google Scholar 

  12. D. V. Georgievskii and M. V. Shamolin, “Sessions of the Workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, “Urgent Problems of Geometry and Mechanics” Named after V. V. Trofimov,” J. Math. Sci., 187, No. 3, 269–271 (2012).

    Article  MATH  Google Scholar 

  13. D. V. Georgievskii and M. V. Shamolin, “Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies,” J. Math. Sci., 187, No. 3, 280–299 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. I. Gurevich, Theory of Jets of an Ideal Liquid [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  15. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern Geometry. Theory and Applications [in Russian], Nauka, Moscow (1979).

  16. V. V. Kozlov, Qualitative Analysis Methods in Rigid Body Dynamics [In Russian], MGU, Moscow (1980).

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

  20. V. A. Samsonov and M. V. Shamolin, “On the problem of body motion in a resisting medium,” Vestn. MGU, Mat., Mekh., 3, 51–54 (1989).

  21. L. I. Sedov, Continuous Medium Mechanics [in Russian], Vols. 1, 2, Nauka, Moscow (1983–1984).

  22. V. V. Trofimov, “Euler equations on finite-dimensional solvable Lie groups,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 5, 1191–1199 (1980).

  23. V. V. Trofimov and M. V. Shamolin, “Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems,” J. Math. Sci., 180, No. 4, 365–530 (2012).

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Google Scholar 

  26. M. V. Shamolin, “On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 2, 65–68 (1997).

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

    Article  MathSciNet  MATH  Google Scholar 

  28. 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).

    MathSciNet  MATH  Google Scholar 

  29. 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).

    MathSciNet  Google Scholar 

  30. M. V. Shamolin, “Complete integrability of equations for motion of a spatial pendulum in overrunning medium flow,” Vestn. MGU, Ser. 1, Mat., Mekh., 5, 22–28 (2001).

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. M. V. Shamolin, “A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking into account rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005).

    MathSciNet  Google Scholar 

  35. M. V. Shamolin, Mothods of Analysis of Dynamical Systems with Variable Dissipation in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007).

    Google Scholar 

  36. M. V. Shamolin, “Complete integrability of equations of motion for a spatial pendulum in a flowing medium taking account of rotational derivatives of moments of its action force,” Izv. Ross Akad. Nauk, Mekh. Tverd. Tela, 3, 187–192 (2007).

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. M. V. Shamolin, “Integrability of some classes of dynamical systems in terms of elementary functions,” Vestn. MGU, Ser. 1, Mat., Mekh., 3, 43–49 (2008).

  41. M. V. Shamolin, “Classification of complete integrability cases in four-dimensional symmetric rigid-body dynamics in a nonconservative field,” in: Contemporary Mathematics and Its Applications [in Russian], 65, Mathematical Physics, Combinatorics, and Optimal Control (2009), pp. 132–142.

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

    MATH  Google Scholar 

  43. M. V. Shamolin, “On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Contemporary Mathematics and Its Applications. [in Russian], 62, Geometry and Mechanics (2009), pp. 131–171.

  44. 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).

    MATH  Google Scholar 

  45. 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).

    Article  Google Scholar 

  46. 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 

  47. M. V. Shamolin, “New case of complete integrability of the dynamical equations on the tangential stratification of three-dimensional sphere,” in: Vestnik SamGU. Estestvennonauchn. Ser., No. 5 (86), 187–189, (2011).

  48. 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 

  49. 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 

  50. 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 

  51. M. V. Shamolin, “Complete list of first integrals of dynamical equations of the spatial motion of a rigid body in a resisting medium under assumption of linear damping,” Vestn. MGU, Ser. 1, Mat., Mekh., 4, 44–47 (2012).

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

  53. M. V. Shamolin, “Comparison of complete integrability cases in dynamics of a two-, three-, and four-dimensional rigid body in a nonconservative field,” in: Contemporary Mathematics and Its Applications [in Russian], 76, Geometry and Mechanics (2012), pp. 84–99.

  54. 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).

    Article  MATH  Google Scholar 

  55. 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).

    Article  MATH  Google Scholar 

  56. 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).

    Article  MathSciNet  MATH  Google Scholar 

  57. 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).

    Article  MathSciNet  MATH  Google Scholar 

  58. 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).

    Article  MathSciNet  MATH  Google Scholar 

  59. 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, Dec. 17–20, 2007 Tech. Univ. Lodz (2007), Vol. 1, pp. 415–422.

  60. 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, Dec. 7–10, 2009, Tech. Univ. Lodz (2009), pp. 91–104.

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

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

  63. 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).

    Article  Google Scholar 

  64. 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, Spain (2011).

  65. 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, Dec. 5–8, 2011, Tech. Univ. Lodz (2011), pp. 11–24.

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

  67. 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).

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 98, Geometry and Mechanics, 2015.

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Shamolin, M.V. New Cases of Integrability of Equations of Motion of a Rigid Body in the n-Dimensional Space. J Math Sci 221, 205–259 (2017). https://doi.org/10.1007/s10958-017-3227-5

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