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Linear systems with quadratic integral and symplectic geometry of Artin spaces. (Russian, English) Zbl 1086.37028
Prikl. Mat. Mekh. 68, No. 3, 371-383 (2004); translation in J. Appl. Math. Mech. 68, No. 3, 329-340 (2004).
The author considers systems of linear differential equations with constant coefficients that admit a nondegenerate quadratic first integral. It was shown by the author [J. Appl. Math. Mech. 56, 803–809 (1992; Zbl 0792.70014)] that such systems can be written in Hamiltonian form with respect to a particular symplectic structure. In the current work, the author exploits this fact further to establish relations between the spectrum of the linear systems and the indices of inertia of the quadratic form and to find conditions under which the singular planes of the quadratic form are Lagrangian with respect to the particular symplectic structure. These results are applied to study the stability of a gyroscopic system.
MSC:
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
34A30 Linear ordinary differential equations and systems
34D05 Asymptotic properties of solutions to ordinary differential equations
37C75 Stability theory for smooth dynamical systems
53D05 Symplectic manifolds (general theory)
70J25 Stability for problems in linear vibration theory
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References:
[1] Kozlov, V.V., Linear systems with a quadratic integral, Prikl. mat. mekh, 56, 6, 900-906, (1992)
[2] Arnol’d, V.I., The conditions for non-linear stability of plane steady curvilinear flows of an ideal fluid, Dokl. akad. nauk SSSR, 162, 5, 975-978, (1995)
[3] Berger, M., Géometrie, (1977), CEDIC
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[5] Williamson, J., On an algebraic problem, concerning the normal forms of linear dynamical systems, Am. J. math., 58, 1, 141-163, (1936) · JFM 63.1290.01
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