The instability of the equilibrium of an inhomogeneous fluid in cases when the potential energy is not minimal

The possibility of extending the methods of proof of instability /1-3/ to the hydrodynamics of an ideal incompressible density-inhomogeneous (stratified) fluid is explored. As distinct from the general statement /3/, the rigid walls of the vessel containing the fluid are assumed to be fixed, so that the purely hydrodynamic part of the problem is isolated. Examples of a two-layer (with and without surface tension) and of a continuously stratified fluid are studied. The main result is to find Lyapunov functionals W which in all cases are increasing, by virtue of the linearized equations of motion of the fluid. The structure of these functionals is such that their growth implies instability in the sense of an increase of the integrals of the disturbance-squared of the hydrodynamic fields (instability in the linear approximation in the mean square). The form of the functionals W is determined by the Hamiltonian statement of the theorem on the instability of finite-dimensional mechanical systems /2/ and by the usual ways of introducing the canonical variables into the hydrodynamic problem /4, 5/. In view of the well-known equivalence of stratification and rotation effects /6, 7/, all the present results hold for two classes of rotating flows of homogeneous fluid. Lyapunov's and Chetayev's theorems (the converse of Lagrange's theorems) are well-known in analytical mechanics; they consist in proving the instability of the equilibrium position of a mechanical system when its potential energy has a maximum or a saddle point /1, 2/. The extension of these theorems to systems that contain rigid bodies and fluid is described in /3/ (Theorem III, p.178).