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Equilibria and Oscillations in a Reversible Mechanical System

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Abstract

This paper investigates symmetric periodic motions (SPM) of reversible mechanical systems. A solution is given to the problem of bilateral continuation of a nondegenerate SPM to the global family of such SPMs. The result is applied to the general case of the Euler problem for a heavy rigid body, when the body parameters are not constrained by equality conditions. Two families of pendulum oscillations are found connecting the lower and upper equilibria.

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REFERENCES

  1. A. A. Zevin, “Nonlocal generalization of Lyapunov theorem,” Nonlinear Anal., Theory, Methods Appl. 28, 1499–1507 (1997).

    MATH  Google Scholar 

  2. A. A. Zevin, “ Global continuation of Lyapunov centre orbits in Hamiltonian systems,” Nonlinearity 12, 1339–1349 (1999).

    Article  MathSciNet  Google Scholar 

  3. A. A. Tikhonov and V. N. Tkhai, “Symmetrical oscillations in the problem of gyrostat attitude motion in a weak elliptical orbit in gravitational and magnetic fields,” Vestn. St. Petersburg Univ.: Math. 48, 119–125 (2015).

    Article  Google Scholar 

  4. V. N. Tkhai, “Lyapunov families of periodic motions in a reversible system,” J. Appl. Math. Mech. 64, 41–52 (2000).

    Article  MathSciNet  Google Scholar 

  5. V. N. Tkhai, “A family of oscillations that connects equilibria,” in Proc. 15th Int. Conf. on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conf.), Moscow, Russia, June 3–5, 2020 (IEEE, Piscataway, N.J., 2020).

  6. V. N. Tkhai, “The behaviour of the period of symmetrical periodic motions,” J. Appl. Math. Mech. 76, 446–450 (2012).

    Article  MathSciNet  Google Scholar 

  7. F. B. Fuller, “An index of fixed point type for periodic orbits,” Am. J. Math. 89 (1), 133–148 (1967).

    Article  MathSciNet  Google Scholar 

  8. B. K. Mlodzievskii, “On the permanent axes in the motion of a heavy rigid body near a fixed point,” Tr. Otd. Fiz. Nauk O-va. Lyubit. Estestvozn., Antropol. Etnogr. 7 (1), 46–48 (1894).

    Google Scholar 

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Funding

This work is supported by the Russian Foundation for Basic Research, project no. 19-01-00146.

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Authors and Affiliations

  1. Institute of Control Sciences of the Russian Academy of Sciences, 117997, Moscow, Russia

    V. N. Tkhai

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  1. V. N. Tkhai
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Correspondence to V. N. Tkhai.

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The cite this work: Tkhai V.N. “Equilibria and Oscillations in a Reversible Mechanical System.” Vestnik of St. Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8 (66), no. 4, pp. 709–715. (In Russian.) https://doi.org/10.21638/spbu01.2021.416.

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Translated by L. Trubitsyna

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Tkhai, V.N. Equilibria and Oscillations in a Reversible Mechanical System. Vestnik St.Petersb. Univ.Math. 54, 447–451 (2021). https://doi.org/10.1134/S1063454121040191

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  • DOI: https://doi.org/10.1134/S1063454121040191

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