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

This article is cited in 11 scientific papers (total in 11 papers)

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


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

Bibliographic databases:

UDC: 517.925.51+531.36
Received: 24.06.2005

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
\by V.~V.~Kozlov
\paper Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization
\jour Funktsional. Anal. i Prilozhen.
\yr 2005
\vol 39
\issue 4
\pages 32--47
\jour Funct. Anal. Appl.
\yr 2005
\vol 39
\issue 4
\pages 271--283

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    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  mathnet  crossref  mathscinet  zmath  elib  elib
    2. Kirillov O.N., “Gyroscopic stabilization in the presence of nonconservative forces”, Dokl. Math., 76:2 (2007), 780–785  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Shavarovskii B.Z., “Classes of factorable and unfactorable polynomial matrices and their relative positions”, Dokl. Math., 75:2 (2007), 318–321  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  mathscinet  elib
    5. E. V. Radkevich, “Matrix Equations and the Chapman–Enskog Projection”, Proc. Steklov Inst. Math., 261 (2008), 229–236  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    6. Palin V.V., Radkevich E.V., “Hyperbolic regularizations of conservation laws”, Russ. J. Math. Phys., 15:3 (2008), 343–363  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. V. V. Palin, “Solvability of matrix Riccati equations”, J. Math. Sci. (N. Y.), 163:2 (2009), 176–187  mathnet  crossref  mathscinet  zmath  elib
    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  mathnet  elib
    9. Chorianopoulos Ch., Lancaster P., “An Inverse Problem For Gyroscopic Systems”, Linear Alg. Appl., 465:SI (2015), 188–203  crossref  mathscinet  zmath  isi  scopus
    10. Valery V. Kozlov, “Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability”, Regul. Chaotic Dyn., 23:1 (2018), 26–46  mathnet  crossref  mathscinet
    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  crossref  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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