Geodesic
Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 32-47

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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. 
@article{FAA_2005_39_4_a2,
     author = {V. V. Kozlov},
     title = {Restrictions of {Quadratic} {Forms} to {Lagrangian} {Planes,} {Quadratic} {Matrix} {Equations,} and {Gyroscopic} {Stabilization}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {32--47},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a2/}
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V. V. Kozlov. Restrictions of Quadratic Forms to Lagrangian Planes, Quadratic Matrix Equations, and Gyroscopic Stabilization. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 32-47. http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a2/