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On the instability of equilibria of conservative systems under typical degenerations. (English. Russian original) Zbl 1189.34097
Differ. Equ. 44, No. 8, 1064-1071 (2008); translation from Differ. Uravn. 44, No. 8, 1033-1040(2008).
Summary: We study systems of differential equations admitting first integrals with degenerate critical points. We find conditions for the instability of equilibria for the cases in which the first integral loses the minimum property. Results of general nature are used in the proof of the impossibility of gyroscopic stabilization of equilibria in conservative mechanical systems under simple typical bifurcations.

MSC:
34D20 Stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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[1] Kozlov, V.V., Prikl. Mat. Mekh., 1992, vol. 56, no. 6, pp. 900–906.
[2] Kozlov, V.V. and Karapetyan, A.A., Differ. Uravn., 2005, vol. 41, no. 2, pp. 186–192.
[3] Poston, T. and Stewart, I., Catastrophe Theory and Its Applications, London: Pitman, 1978. Translated under the title Teoriya katastrof i ee prilozheniya, Moscow: Mir, 1980. · Zbl 0548.58007
[4] Chetaev, N.G., Ustoichivost’ dvizheniya (Stability of Motion), Moscow: Gostekhizdat, 1955.
[5] Koiter, W., Proc. Kon. ned. acad. wet., 1965, vol. 68, no. 3, pp. 107–113.
[6] Kozlov, V.V., Prikl. Mat. Mekh., 1986, vol. 50, no. 6, pp. 928–937.
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