Variation Principles of Hydrodynamics

It is shown that the Lagrangian equations for the motion of both incompressible and compressible fluids can be derived from variation principles. As has been pointed out by C. C. Lin, an important feature of these principles is the boundary condition: The coordinates of each particle (and not merely the normal component of its velocity) must be specified. A systematic application of known results from the calculus of variations reveals new interrelations between such hydrodynamic results as Bernoulli's principle and the circulation theorem. Their derivation is both simplified and systematized. Clebsch's transformation is found to have an important relation to the problem of integrating the vorticity equation. The general solution of this problem for nonbarotropic flow is obtained. This reduces the second-order Lagrange equations to a set of first-order equations, in which the potential of the irrotational component replaces the pressure. The entropy gradients and a remarkable quantity, defined as the time-integral of the temperature of a particle of the fluid, also appear.