zbMATH — the first resource for mathematics

Restrictions of quadratic forms to Lagrangian planes, quadratic matrix equations, and gyroscopic stabilization. (English. Russian original) Zbl 1113.37040
Funct. Anal. Appl. 39, No. 4, 271-283 (2005); translation from Funkts. Anal. Prilozh. 39, No. 4, 32-47 (2005).
Summary: We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70J25 Stability for problems in linear vibration theory
70J10 Modal analysis in linear vibration theory
70H05 Hamilton’s equations
Full Text: DOI Link
[1] V. V. Kozlov, General Theory of Vortices, Springer-Verlag, 2003. · Zbl 1104.37050
[2] J. Williamson, ”On an algebraic problem concerning the normal forms of linear dynamical systems,” Amer. J. Math., 58, No.1, 141–163 (1936). · Zbl 0013.28401
[3] F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow, 1967. · Zbl 0050.24804
[4] V. V. Kozlov and A. A. Karapetyan, ”On the stability degree,” Differentsial’nye Uravneniya, 41, No.2, 186–192 (2005); English transl.: Differential Equations, 41, No. 2, 195–201 (2005).
[5] V. V. Kozlov, ”Linear systems with a quadratic integral,” Prikl. Mat. Mekh., 56, No.6, 900–906 (1992); English transl.: J. Appl. Math. Mech., 56, No. 6, 803–809 (1992).
[6] S. I. Gelfand, ”On the number of solutions of a quadratic equation,” in: GLOBUS, IUM General Seminar [in Russian], No. 1, MCCME, Moscow, 2004, pp. 124–133.
[7] V. V. Kozlov, ”The spectrum of a linear Hamiltonian system and the symplectic geometry of the complex Artin space,” Dokl. Ross. Akad. Nauk, 393, No.24, 453–455 (2003).
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.