Variational methods in the hydrodynamic theory of liquidHe4

This paper presents a derivation of the dissipationless two-fluid equations of motion for liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ based on a variational procedure that is closely related to one formulated originally by Zilsel. The Lagrangian that appears in this treatment has been derived before by applying statistical-mechanical methods to an elementary-excitation description of the flowing liquid; the main steps are recounted here. An important new development is that for uniform flow, the Lagrangian is shown to be a Legendre transform of the internal energy. It is a particular thermodynamic potential for which the primary, independent variables are clearly exhibited. Identification of these variables makes it possible to avoid certain steps in Zilsel's procedure which have been criticized by several workers, while one arrives at the same equations of motion. Furthermore, it is shown, by example, that the Lagrangian density postulated by Zilsel is exactly the same as that assumed by Lhuillier, Francois, and Karatchentzeff (LFK), but that terms have been grouped differently in the two treatments. This should eliminate apprehension expressed by LFK about the reliability of Zilsel's Lagrangian. The results derived here also bring new unity to the work of Zilsel and its extension by Jackson, and certain work of Khalatnikov and its extensions by LFK and by Geurst. Finally, a discussion is given of a proposal made by Lin for modifying Zilsel's variational treatment of liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$.