HAMILTONIAN-FORMALISM FOR NONLINEAR-WAVES

The Hamiltonian description of hydrodynamic type systems for applicati on to plasmas, hydrodynamics, and magnetohydrodynamics is reviewed wit h emphasis on the problem of introducing canonical variables. The rela tion to other Hamiltonian approaches, in particular natural-variable P oisson's brackets, is pointed out. It is shown that the degeneracy of noncanonical Poisson's brackets relates to the special type of symmetr y, the relabeling transformations of fluid-particle Lagrangian markers , from which all known vorticity conservation theorems, such as Ertel' s, Cauchy's, Kelvin's, as well as the vorticity frozenness and the top ological Hopf invariant, derive. The application of canonical variable s to collisionless plasma kinetics is described. The Hamiltonian struc ture of Benney's equations and of the Rossby wave equation is discusse d. Davey-Stuartson's equation is given the Hamiltonian form. A general method for treating weakly nonlinear waves is presented based on clas sical perturbation theory and the Hamiltonian reduction technique.