A new Hamiltonian formulation for fluids and plasmas. Part 1. The perfect fluid

摘要

A new formulation of the Hamiltonian structure underlying the perfect fluid Equations is presented. Besides time, a parameter e is also used. Correspondingly, there are two interdependent systems of equations expressing time evolution and e evolution respectively, The accessibility equations define the e dynamics and give the variation in the usual Eulerian fluid variables as determined by the generating functions. The time evolutions of both the jtolerian fluid variables and the generating functions are obtained from an action principle. The consistency of the e and the time dynamics is crucial for his formulation, i.e. the accessibility equations must be propagated in time by Euler-Lagrange equations. The reason for introducing this new formulation fits power in certain applications where the existing Hamiltonian alternatives &em less convenient to use. In particular, it is a promising tool for Hamiitonian prturbation theory. We consider the small-amplitude expansion, and find spry simply and naturally, the Hermitian structure of the linearized ideal fluid filiations as well as coupling coefficients for resonant three-wave interaction exhibiting the Manley-Rowe relations.