Equilibria and oscillations in a reversible mechanical system
DOI:
https://doi.org/10.21638/spbu01.2021.416Abstract
The 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 Euler’s problem on a heavy rigid body, when the body parameters are not bound by equality conditions and two families of pendulum oscillations are found connecting the lower and upper equilibria.Keywords:
reversible mechanical system, equilibrium, symmetric periodic motion, continuation, global family, Euler’s problem, pendulum oscillations
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References
Литература
1. Zevin A.A. Nonlocal generalization of Lyapunov theorem. Nonlinear Analysis, Theory, Methods and Applications 28 (9), 1499–1507 (1997).
2. Zevin A.A. Global continuation of Lyapunov centre orbits in Hamiltonian systems. Nonlinearity 12, 1339–1349 (1999).
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References
1. Zevin A.A. Nonlocal generalization of Lyapunov theorem. Nonlinear Analysis, Theory, Methods and Applications 28 (9), 1499–1507 (1997).
2. Zevin A.A. Global continuation of Lyapunov centre orbits in Hamiltonian systems. Nonlinearity 12, 1339–1349 (1999).
3. Tikhonov A.A., Tkhai V.N. Symmetrical Oscillations in the Problem of Gyrostat Attitude Motion in a Weak Elliptical Orbit in Gravitational and Magnetic Fields. Vestnik of Saint Petersburg university. Series 1. Mathematics. Mechanics. Astronomy, iss. 2, 279–287 (2015). (In Russian) [Engl. transl.: Vestnik St. Petersburg University. Mathematics 48 (2), 119–125 (2015)].
4. Tkhai V.N. Lyapunov families of periodic motions in a reversible system. Prikladnaya matematika i mekhanika 64, iss. 1, 56–72 (2000). (In Russian) [Engl. transl.: Journal Applied Mathematics and Mechanics 64, iss. 1, 41–52 (2000)].
5. Tkhai V.N. A family of oscillations that connects equilibria. 2020 15th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy’s Conference). IEEE Xplore (2020).
6. Tkhai V.N. On the behavior of the period of symmetric periodic movements. Prikladnaya matematika i mekhanika 76, вып. 4, 616–622 (2012). (In Russian)
7. Fuller F.B. An index of fixed point type for periodic orbits. American Journal of Mathematics 89 (1), 133–148 (1967).
8. Mlodzievskii B.K. On the permanent axes in the motion of a heavy rigid body near a fixed point. In: Trudi otdeleniya fizicheskih nauk obshestva lubitelei estestvovaniya, antropologii i etnografii. Vol. 7, iss. 1, 46–48 (1894). (In Russian)
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Published
2022-01-04
How to Cite
Tkhai, V. N. (2022). Equilibria and oscillations in a reversible mechanical system. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(4), 709–715. https://doi.org/10.21638/spbu01.2021.416
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Mechanics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
