# zbMATH — the first resource for mathematics

Gyroscopic stabilization and parametric resonance. (English. Russian original) Zbl 1038.70014
J. Appl. Math. Mech. 65, No. 5, 715-721 (2001); translation from Prikl. Mat. Mekh. 65, No. 5, 739-745 (2001).
The author studies small oscillations of a mechanical system near the equilibrium state which are described by the equations $$\ddot x + \Gamma\dot x +Px = 0$$, $$x\in \mathbb R^n$$, where $$\Gamma$$ is a skew-symmetric $$n\times n$$-matrix, while the matrix $$P$$ is symmetric. New sufficient conditions are established for gyroscopic stabilization of unstable equilibria by means of gyroscopic forces with a degenerate matrix.

##### MSC:
 70J25 Stability for problems in linear vibration theory 70J40 Parametric resonances in linear vibration theory
##### References:
 [1] Chetayev, N. G.: The stability of motion. Papers on analytical mechanics.. (1962) [2] Bulatovič, R. M.: The stability of linear potential gyroscopic systems in cases when the potential energy has a maximum. Prikl. mat. Mekh. 61, No. 3, 385-389 (1997) · Zbl 0884.70009 [3] Huseyn, K.; Hagedorn, P.; Tescher, W.: On the stability of linear conservative gyroscopic systems. Zamp 34, 807-815 (1983) · Zbl 0538.70022 [4] Bolotin, S. V.; Kozlov, V. V.: Asymptotic solutions of the equations of dynamics. Matematika, mekhanika 6, 98-103 (1983) [5] Pozharitskii, G. K.: The unsteady motion of conservative holonomic systems. Prikl. mat. Mekh. 20, No. 1, 429-433 (1956) [6] Hill, G. W.: On the part of the motion of the lunar perigee which is a function of the mean motion of the Sun and Moon. Acta math. 8, 1-36 (1886) · JFM 18.1106.01 [7] Bolotin, S. V.: The Hill determinant of a periodic trajectory. Vestnik MGU. Ser. 1, matematika, mekhanika 3, 30-34 (1988) [8] Kozlov, V. V.: The stabilization of the unstable equilibria of charges by intense magnetic fields. Prikl. mat. Mekh. 61, No. 3, 390-397 (1997) · Zbl 0881.93064 [9] Yakubovich, V. A.; Starzhinskii, V. M.: Linear differential equations with periodic coefficients and their applications. (1972) [10] Zevin, A. A.: Some conditions for the existence and stability of periodic oscillations in non-linear non-autonomous Hamilton systems. Prikl. mat. Mekh. 48, No. 4, 637-646 (1984) · Zbl 0585.70021 [11] Krein, M. G.: The criteria of stable boundedness of the solutions of periodic canonical systems. Prikl. mat. Mekh. 19, No. 6, 641-680 (1955)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.