A hydrodynamic analogy of the solution to double-averaged Hill problem

A hydrodynamic analogy for the solution of doubly averaged Hill problem obtained by M.L. Lidov [1] is discussed. S.A. Chaplygin used a similar analogy reducing the two-dimensional problem of motion of a compressible fluid to the same problem for some fictitious incompressible fluid [2]. In this case, we are led to the range of ideas which go back to the work by N.E. Zhukowski [3]. Two versions of this hydrodynamic analogy, on the basis of the model of a stratified fluid (exact analogy) and on the basis of the model of a homogeneous fluid (approximate analogy), are considered as well as some consequences of them.