Inversion of lagrange's theorem for a rigid body with a cavity containing a viscous liquid

The stability of the state of equilibrium of a rigid body with a cavity partly or completely filled with a viscous incompressible liquid possessing surface tension is cosidered in a linear form. Lyapunov's direct method is used to show that the system is unstable if the second variation of the potential energy can take negative values. A priori lower and upper bounds for the solutions, when the perturbations are increased, are obtained. The lower bound guarantees exponential growth of the deviations of the solid and liquid particles from the equilibrium state. The upper bound shows that the solutions cannot increase at more than an exponential rate. In both cases the exponents are calculated from the parameters of the equilibrium state and the initial data for the perturbation fields.