Operator Approach to the Problem of Small Oscillations of a Pendulum with a Cavity Partially Filled with a Heavy Viscous Fluid

We provide the new operator approach to the linearized problem of small oscillations of a solid body (pendulum) with a cavity partially filled with a heavy viscous fluid. We use methods of the general operator theory for operators and operator-matrices acting in Hilbert space and in space with indefinite metrics (Krien space). In this way, we obtain operator evolution problem and formulate theorem on its solvability. Corresponding spectral problem is studied in detail. We connect known approach based on considering the operator pencil of S.G. Krein with the operator-matrix approach. We prove theorem on the recalculation of eigen and associated elements, provide theorem on J-orthonormal or almost J-orthonormal basis properties. We found also the decomposition of the evolution problem solution into a series of eigenvectors.