The Instability Index Formula. Application to the Sobolev problem on a rotating top filled with a viscous fluid

Many linear equations in elasticity and hydrodynamics can be written in the form
$$ A(u) = F\ddot u +(D+iG)\dot u +Tu =0, \quad u= u(t), $$
where the dots denote the differentiation with respect to time $t$ and $F,D,G,T$ are the symmetric operators on a suitable Hilbert space $H$ (the last three operators are responsible for the dissipative, gyroscopic and potential forces). The Laplace symbol of this equation coincides with the quadratic operator pencil
$$ A(\lambda) = F\lambda^2 +(D + i G)\lambda + T. $$
Assuming that the inertia indices $\nu(F)$ and $\nu(T)$ of the operators $F$ and $T$ are finite we show that the number $\kappa(A)$ of the eigenvalues of the pencil $A(\lambda)$ in the right-half plane is expressed by the formula
$$ \kappa(A) = \nu(F) + \nu(T) - \varepsilon^+(A), $$
where $\varepsilon^+(A)$ is the sum of the lengths of the Jordan chains corresponding to the imaginary eigenvalues with the positive sign characteristics. In particular, we always have $\kappa(A) \leqslant \nu(F) + \nu (T)$. We will tell on the history of this problem and show that this formula allows us to obtain a stability criterion in the Sobolev problem on a rotating top filled with a viscous liquid.