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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.

MSC:
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
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References:
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