Hydrodynamics for an Ideal Fluid: Hamiltonian Formalism and Liouville-equation

Clebsch 1'2) was the first to derive, in 1859, the hydrodynamic equations for an ideal fluid from a variational principle for Euler coordinates. His derivation was, however, restricted to the case of an incompressible fluid. Later Bateman 3) showed that the analysis of Clebsch also applies to compressible fluids if the pressure is a function of density alone. Finally, in 1968, Seliger and Whitham 4) formulated a Lagrangian density for the most general case, i.e. taking also into account the dependence on entropy. From a Lagrangian formalism it is of course in general possible to go over to a Hamiltonian description. For hydrodynamics, this was done by Kronig and Thellung 5'6) in order to quantize the fluid equations. As they based their work on Bateman's analysis, their results only apply to the case of isentropic (or, alternatively, isothermal) flow. Recently, there has been renewed interest in a Hamiltonian formulation of hydrodynamics. In an interesting paper Enz and Turski 7) considered hydrodynamic fluctuations on the basis, and with the limitations, of the formalism developed by Thellung6). In this paper we will develop a Hamiltonian formalism for the general case of Seliger and Whitham and discuss a number of statistical properties of an ideal fluid. This discussion will enable us to study in a subsequent paper nonlinear fluctuations in a real fluid. In section 2 we discuss the Clebsch representat ion of the fluid velocity field. Seliger and Whitham's 4) variational principle, which is based on this representation, is reviewed in section 3. We then introduce a Hamiltonian description of hydrodynamics and define Poisson-brackets in terms of the