Analogues of the lagrange theorem in the hydrodynamics of whirling and stratified liquids

Abstract

Assertions of the type of Lagrange's theorem are presented for three new classes of flows of an ideal incompressible liquid. The rest states of a liquid which is inhomogeneous with respect to its density (continuously stratified) and located in an external field of mass forces comprise the first class. Certain whirling (rotating) flows of a liquid which is homogeneous with respect to its density belong to the second and third classes. Unlike the situations which have been studied previously, flows belonging to the second and third classes are not states of relative or absolute rest and do not possess free boundaries. At the same time the formulations and proofs of the assertions are practically repeats of one another for all three cases. The question of the existence of an analogue of Lagrange's theorem in hydrodynamics has been studied in a number of papers (/1–4/, etc.).