Geodesic
Invariant Planes, Indices of Inertia, and Degrees of Stability of Linear Dynamic Equations
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 1, Tome 258 (2007), pp. 154-161.

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Spectral properties of linear dynamic equations linearized at equilibrium points are analyzed. The analysis involves a search for invariant planes that are uniquely projected onto the configuration plane. In turn, the latter problem reduces to the solution of a quadratic matrix equation of special form. Under certain conditions, the existence of two different solutions is proved by the contraction mapping method. An estimate for the degree of stability is obtained in terms of the index of inertia of potential energy. 
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V. V. Kozlov. Invariant Planes, Indices of Inertia, and Degrees of Stability of Linear Dynamic Equations. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 1, Tome 258 (2007), pp. 154-161. http://geodesic.mathdoc.fr/item/TM_2007_258_a10/

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