Variational Principle in Hydrodynamics

Abstract

A canonical formalism in hydrodynamics -is developed by aid of Clebsch's transformation. If we consider the adiabatic process where the entropy is conserved, vortex motion is found to be closely connected with the entropy. Euler's hydrodynamical equation emerges fllctorised into four fundamental equations which are discussed from the thermodynamical point of view. Many important relations in hydrodynamics are naturally introduced by these equations. Landau's theory of quantum hydrodynamics is criticized upon our theory. The variational method or canonical formalism has not so much been developed compared with the study of the motion of liquid itself since it seems to bear only academic interests. The reason why the author pays his attention to this problem is to criticize the possibility of quantum hydrodynamics developed by LandauY According to the current theory of quantum electrodynamics, the canonical formalism of the classical part of hydrodynamics as a field theory is necessary for the purpose of quantization. The most important field quantity in hydrodynamics is the velocity field which is generally composed of two parts, namely gradient field and rotational field. From this point of view, Clebsch2) has developed skilful representation of the velocity field of which an account can be found in Lamb's Hydrodynamics. Although the Hamilton form of equations established in this boo~, it is presented in the hydrodynamics of substantial form and it seems inconvenient to use it for our field point of view. But it is not difficult to rewrite it using the Clebsch's transformation as will be discussed in § 2 and § 3. There emerge many interesting points in this formalism if we take into account the thermodynamical consideration. Of course, we must limit our- considerations in the framework of usual variational principles to the reversible processes, in particular to adiabatic process, where the entropy is conserved. One will find in § 4 that the formalism of § 3 does not change when the geometrical parameter A which determines the vortex lines is interpreted as the entropy S. In both cases, the canonical equations will be combined into one equation, that is, Euler's equation of motion where the geometrical parameters or thermodynamical quantities all disappear. The canonical equations themselvt;s have their own physical mean­ ings and many wellknown relations in hydrodynamis are easily derived from them in § 5. In addition to this, the isothermal reversible process is discussed and we have the similar equations where some thermodynamical functions are correspondingly transformed. In §- 6, 117