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.

Reviewer: Dragomir L. Dragnev (MR2092153)

##### 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 |

##### Keywords:

linear systems; Artinian space; systems with gyroscopic forces; linear differential equations; nondegenerate quadratic first integral; Hamiltonian form; symplectic structure; stability; gyroscopic system; Lagrangian form
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\textit{V. V. Kozlov}, Prikl. Mat. Mekh. 68, No. 3, 371--383 (2004; Zbl 1086.37028); translation in J. Appl. Math. Mech. 68, No. 3, 329--340 (2004)

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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 |

[4] | Kozlov, V.V., The general theory of vortices, (1998), Izd. Dom “Udmurtskii Universitet” Paris |

[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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