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

Abstract

A new formulation of the Hamiltonian structure underlying the perfect fluid equations is presented. Besides time, a parameter c 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 Eulerian fluid variables and the generating functions are obtained from an action principle. The consistency of the e and the time dynamics is crucial for this formulation, i.e. the accessibility equations must be propagated in time by the Euler–Lagrange equations. The reason for introducing this new formulation is its power in certain applications where the existing Hamiltonian alternatives seem less convenient to use. In particular, it is a promising tool for Hamiltonian perturbation theory. We consider the small-amplitude expansion, and find, very simply and naturally, the Hermitian structure of the linearized ideal fluid equations as well as coupling coefficients for resonant three-wave interaction exhibiting the Manley–Rowe relations.