Some Transformations of the Hydrodynamic Equations

Abstract

The Lagrangian equations of hydrodynamics are transformed to general coordinates. Since they involve two sets of variables (dependent and independent) two transformations are involved. The independent coordinates are usually taken to be the time and the initial positions of the particles of the fluid. Other kinds of independent variables are often useful. If they are chosen so that they describe the motion of a second fluid, the Lagrangian equations become a tool for comparing the motion of the two fluids. If the second fluid obeys the same laws as the first, the comparison is between two modes of motion of the same fluid. The study of this problem leads to a new derivation of the Eulerian equations from the Lagrangian. If the two modes of motion differ only slightly, the perturbation equations are obtained in generalized coordinates. They satisfy a variation principle. This can be transformed so as to exhibit the general form of a pseudo potential energy function that is useful in the study of stability problems.