Abstract
We investigate the stability of an equilibrium state of gyroscopic coupled conservative systems in the case where the force function does not attain a local maximum in this state. We consider the situation where the gyroscopic coupling is weak with respect to a part of coordinates and strong with respect to the other part.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. J. Routh, A Treatise on the Stability of a Given State of Motion, McMilland, London (1892).
W. Thomson and P. Tait, Treatise on Natural Philosophy, Clarendon Press, Oxford (1867).
N. G. Chetaev, Stability of Motion [in Russian], Academy of Sciences of the USSR, Moscow (1962).
A. V. Karapetyan and V. V. Rumyantsev, “Stability of conservative and dissipative systems,” in: VINITI Series in General Mechanics [in Russian], Vol. 6, VINITI, Moscow (1983).
V. V. Rumyantsev and S. P. Sosnitskii, “On the instability of an equilibrium state of holonomic conservative systems,” Prikl.Mat.Mekh., 57, Issue 6, 144–166 (1993).
P. Hagedorn, “Ñber die Instabilität conservativer Systeme mit gyroscopischen Kräften,” Arch.Ration.Mech.Anal., 58, No. 1, 1–9 (1975).
S. V. Bolotin and V. V. Kozlov, “On asymptotic solutions of equations of dynamics,” Vestn.Mosk.Univ., Ser.Mat.Mekh., No. 4, 84–89 (1980).
S. P. Sosnitskii, “Hamiltonian action and stability of an equilibrium state of conservative systems,” in: Problems of Dynamics and Stability of Multidimensional Systems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1991), pp. 99–106.
S. V. Bolotin and P. Negrini, “Asymptotic trajectories of gyroscopic systems,” Vestn.Mosk.Univ., Ser.Mat.Mekh., No. 6, 66–75 (1993).
S. Bolotin and P. Negrini, “Asymptotic solutions of Lagrangian systems with gyroscopic forces,” Nonlin.Different.Equat.Appl., 2, No. 4, 417–444 (1995).
S. P. Sosnitskii, “On the stabilization of an equilibrium state of conservative systems using gyroscopic forces,” Prikl.Mat.Mekh., 64, Issue 1, 59–69 (2000).
S. P. Sosnitskii, “On the instability of equilibrium of gyroscopic coupled conservative systems,” Int.J.Nonlin.Mech., 35, No. 3, 487–496 (2000).
V. V. Kozlov, “Asymptotic solutions of equations of classical mechanics,” Prikl.Mat.Mekh., 46, Issue 4, 573–577 (1982).
V. Moauro and P. Negrini, “On the inversion of the Lagrange–Dirichlet theorem,” Different.Integr.Equat., 2, No. 4, 471–478 (1989).
C. Briot and T. Bouquet, “Recherches sur les propriètés des fonctions définies par des équations différentielles,” J.École Polytechn.Paris, 21, No. 36, 133–199 (1856).
G. V. Kamenkov, Selected Works [in Russian], Nauka, Moscow (1971–1972).
V. I. Zubov, Stability of Motion [in Russian], Vysshaya Shkola, Moscow (1973).
S. P. Sosnitskii, “On some cases of instability of an equilibrium state of natural systems,” Ukr.Mat.Zh., 37, No. 1, 124–127 (1985).
N. A. Kil'chevskii, A Course of Theoretical Mechanics [in Russian], Nauka, Moscow (1977).
Yu. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems [in Russian], Nauka, Moscow (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sosnyts'kyi, S.P. On the Stability of an Equilibrium State of Gyroscopic Coupled Systems. Ukrainian Mathematical Journal 55, 311–321 (2003). https://doi.org/10.1023/A:1025420530338
Issue Date:
DOI: https://doi.org/10.1023/A:1025420530338