A Canonical Transformation Relating the Lagrangian and Eulerian Description of Ideal Hydrodynamics

In fluid mechanics, two ways exist to specify the fields. The one most often used is the Eulerian description, in which the fields are considered as functions of the position in space, and time. The Lagrangian description, on the other hand, is based on the observation that many quantities specifying the fluid refer more fundamental ly to small identifiable pieces of matter, the "fluid particles". In the Lagrangian description one therefore considers a~ll the fields as functions of the time and the label of the fluid particle, to which they pertain. As the fluid equations for ideal flow in the Lagrangian specification reflect quite closely the equations of motion for ordinary point particles, it is not surprising that a variational principle for these equations is already known since and resembles the one for point-particles. In fact, the formulation of this variational principle is due to Lagrange himself. The formulation of a variational principle for the Eulerian way to specify the fields, however, has proceeded in steps and spanned a long period. Clebsch l) was the first to introduce in 1859 a variational principle for the case of incompressible flow. His analysis however applies to other cases of constrained flow as well, such as the case of compressible isentropic flow, which was investigated in detail by Bateman 2) in 1929. It was only in 1968 that Seliger and Whitham extended Clebsch's variational principle to the most general case of compressible, nonisentropic flow. The Lagrangian they introduce is of a somewhat unexpected form; yet the corresponding Hamiltonian is just the energy and is therefore the same as the one found in the variational principle for the