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 Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 4, Pages 32–47 (Mi faa83)

Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization

V. V. Kozlov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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.

Keywords: Hamiltonian function, symplectic structure, quadratic form, Williamson normal form, vortex plane

DOI: https://doi.org/10.4213/faa83

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English version:
Functional Analysis and Its Applications, 2005, 39:4, 271–283

Bibliographic databases:

UDC: 517.925.51+531.36

Citation: V. V. Kozlov, “Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization”, Funktsional. Anal. i Prilozhen., 39:4 (2005), 32–47; Funct. Anal. Appl., 39:4 (2005), 271–283

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa83
• https://doi.org/10.4213/faa83
• http://mi.mathnet.ru/eng/faa/v39/i4/p32

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Kozlov, “Invariant Planes, Indices of Inertia, and Degrees of Stability of Linear Dynamic Equations”, Proc. Steklov Inst. Math., 258 (2007), 147–154
2. Kirillov O.N., “Gyroscopic stabilization in the presence of nonconservative forces”, Dokl. Math., 76:2 (2007), 780–785
3. Shavarovskii B.Z., “Classes of factorable and unfactorable polynomial matrices and their relative positions”, Dokl. Math., 75:2 (2007), 318–321
4. B. Z. Shavarovskii, “Finding a complete set of solutions or proving unsolvability for certain classes of matrix polynomial equations with commuting coefficients”, Comput. Math. Math. Phys., 47:12 (2007), 1902–1911
5. E. V. Radkevich, “Matrix Equations and the Chapman–Enskog Projection”, Proc. Steklov Inst. Math., 261 (2008), 229–236
6. Palin V.V., Radkevich E.V., “Hyperbolic regularizations of conservation laws”, Russ. J. Math. Phys., 15:3 (2008), 343–363
7. V. V. Palin, “Solvability of matrix Riccati equations”, J. Math. Sci. (N. Y.), 163:2 (2009), 176–187
8. B. Z. Shavarovskii, “Faktorizuemost mnogochlennykh matrits vo vzaimosvyazi so stepenyami ikh elementarnykh delitelei”, Zh. vychisl. matem. i matem. fiz., 52:7 (2012), 1171–1184
9. Chorianopoulos Ch., Lancaster P., “An Inverse Problem For Gyroscopic Systems”, Linear Alg. Appl., 465:SI (2015), 188–203
10. Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46
11. Denisova N.V., “Instability Degree and Singular Subspaces of Integral Isotropic Cones of Linear Systems of Differential Equations”, Dokl. Math., 97:1 (2018), 35–37
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