Regular and Chaotic Dynamics

, Volume 13, Issue 3, pp 204–220 | Cite as

Absolute and relative choreographies in rigid body dynamics

  • A. V. Borisov
  • A. A. Kilin
  • I. S. Mamaev
Research Articles


For the classical problem of motion of a rigid body about a fixed point with zero area integral, we present a family of solutions that are periodic in the absolute space. Such solutions are known as choreographies. The family includes the well-known Delone solutions (for the Kovalevskaya case), some particular solutions for the Goryachev-Chaplygin case, and the Steklov solution. The “genealogy” of solutions of the family naturally appearing from the energy continuation and their connection with the Staude rotations are considered. It is shown that if the integral of areas is zero, the solutions are periodic with respect to a coordinate frame that rotates uniformly about the vertical (relative choreographies).

Key words

rigid-body dynamics periodic solutions continuation by a parameter bifurcation 

MSC2000 numbers

76B47 37J35 70E40 


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  1. 1.
    Borisov, A.V. and Mamaev, I.S., Dinamika tverdogo tela (Rigid Body Dynamics), Moscow-Izhevsk: Inst. komp. issled., RCD, 2005.zbMATHGoogle Scholar
  2. 2.
    Gorr, G.V., Kudryashova, L.V., and Stepanova, L.A., Klassicheskie zadachi dinamiki tverdogo tela (Classical Problems in Dynamics of Rigid Bodies), Kiev: Naukova Dumka, 1978.Google Scholar
  3. 3.
    Leimanis, E., The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Berlin: Springer-Verlag, 1965.zbMATHGoogle Scholar
  4. 4.
    Marinbakh, M.A., Ljapunov Periodic Motions of a Heavy Rigid Body with a Fixed Point, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1979, no. 5, pp. 75–79.Google Scholar
  5. 5.
    Marinbakh, M.A., The General Case of Lyapunov’s Periodic Motions of a Heavy Solid Body with a Single Fixed Point, Prikl. Mat. Mekh., 1981, vol. 45, pp. 800–807 [Engl. transl.: J. Appl. Math. Mech., 1982, vol. 45, pp. 598–603].MathSciNetGoogle Scholar
  6. 6.
    Sergeev, V.S., On Periodic Solutions to the Equations of Motion of a Heavy Rigid Body with a Fixed Point, Vestnik Moskov. Univ. Ser. I Mat. Meh., 1969, no. 1, pp. 40–51.Google Scholar
  7. 7.
    Mettler, E., Periodische und asymptotische Bewegungen des unsymmetrischen schweren Kreisels, Math. Z., 1937, vol. 43, no. 1, pp. 59–100.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kozlov, V.V., Metody kachestvennogo analiza v dinamike tverdogo tela (Methods of Qualitative Analysis in the Dynamics of a Rigid Body), Moscow: Moskov. Gos. Univ., 1980. Second edition: Izhevsk-Moscow: SPC Regular & Chaotic Dynamics, 2000.zbMATHGoogle Scholar
  9. 9.
    Arkhangel’ski, Yu.A., Dinamika bystro vraschayuschihsya tevrdyh tel (Dynamics of Rapidly Rotating Rigid Bodies), Moscow: Nauka, 1985.Google Scholar
  10. 10.
    Novikov, S.P., The Hamiltonian Formalism and a Many-Valued Analogue of Morse Theory, Usp. Mat. Nauk, 1982, vol. 37, no. 5, pp. 3–49 [Engl. transl.: Russ. Math. Surv., 1982, vol. 37, no. 5, pp. 1–56].Google Scholar
  11. 11.
    Borisov, A.V. and Simakov, N.N., Period Doubling Bifurcation in Rigid Body Dynamics, Regul Chaotic Dyn., 1997, vol. 2, no. 1, pp. 64–75.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Chaotic Motions and Transition to Stochasticity in the Classical Problem of the Heavy Rigid Body with a Fixed Point, Nuovo Cim., 1981, vol. 61B, no. 1, pp. 1–20.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Chenciner, A. and Montgomery, R., A Remarkable Periodic Solution of the Three Body Problem in the Case of Equal Masses, Ann. Math., 2000, vol. 152, pp. 881–901.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Simó, C., Dynamical Properties of the Figure Eight Solution of the Three-Body Problem, in Celestial Mechanics, Celestial mechanics (Evanston, IL, 1999), vol. 292 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2002, pp. 209–228.Google Scholar
  15. 15.
    Gorr, G.V. and Savchenko, A.Ya., On One Periodic Motion in the Kovalevskaya Solution, Mekh. tverd. tela, 1971, Issue 3, pp. 64–69.Google Scholar
  16. 16.
    Gorr, G.V., On a Certain Motion of a Heavy Solid in the Gorjacev-Caplygin Case, Prikl. Mat. Mekh., 1970, vol. 34, pp. 1139–1143 [Engl. transl.: J. Appl. Math. Mech., 1973, vol. 34, pp. 1075–1080].Google Scholar
  17. 17.
    Gorr, G.V. and Levitskaya, G.D., On One Periodic Motion of the Goryachev-Chaplygin Gyroscope, Mekh. tverd. tela, 1971, Issue 3, pp. 101–106.Google Scholar
  18. 18.
    Steklov, V.A., New Particular Solution to Differential Equations of Motion of a Heavy Rigid Body Having a Fixed Point, Tr. Otd. Fiz. Nauk Obshch. Lyubit. Estestvozn., 1899, vol. 10, no. 1, pp. 1–3.Google Scholar
  19. 19.
    Kuz’min, P.A., An Addition to the V.A. Stekov Case of the Motion of a Heavy Rigid Body About a Fixed Point, Prikl. Mat. Mekh., 1952, vol. 16, no. 3, pp. 243–245.MathSciNetGoogle Scholar
  20. 20.
    Kharlamov, M.P. and Sergeev, E.K., The Construction of the Complete Solution of One Problem of Rigid-Body Dynamics, Mekh. tverd. tela, 1982, Issue 14, pp. 33–38.Google Scholar
  21. 21.
    Barkin, Yu.V., Periodic and Conditionally Periodic Solutions in the Problem of Motion of a Heavy Solid about a Fixed Point, Prikl. Mat. Mekh., 1981, vol. 45, pp. 535–544 [Engl. transl.: J. Appl. Math. Mech., 1982, vol. 45, pp. 391–397].MathSciNetGoogle Scholar
  22. 22.
    Borisov, A.V. and Mamaev, I.S., Euler-Poisson Equations and Integrable Cases, Regul Chaotic Dyn., 2001, vol. 6, pp. 253–274.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Chaplygin, S.A., A New Particular Solution to the Problem on the Rotation of a Heavy Rigid Body about a Fixed Point, in Collected Papers, Moscow-Leningrad: Gostekhizdat, 1948, vol. 1, pp. 125–132.Google Scholar
  24. 24.
    Delone, N.B., On the Geometric Interpretation of Integrals of Rigid-Body Motion About a Fixed Point Given by S.V. Kovalevskaya, Mat. Sb. Kruzhka Lyubit. Mat. Nauk, 1892, vol. 16, no. 2, pp. 346–351.Google Scholar
  25. 25.
    Macmillan, W.D., Dynamics of Rigid Bodies, New York-London: McGraw-Hill, 1936.zbMATHGoogle Scholar
  26. 26.
    Staude, O., Über permanente Rotationaxen bei der Bewegung eines schweren Körpers um einen festen Punkt, J. Reine Angew. Math., 1894, vol. 113, no. 4, pp. 318–334.Google Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

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