Hamiltonian approach to the description of non-linear plasma phenomena

This review discusses the methods of description of non-linear processes. We demonstrate the usefulness of the Hamiltonian approach to these problems. We show the existence of Hamiltonian structures for a number of plasma situations. The choice of normal variables results in a standard form of equations for all kinds of problems. The actual physics involved changes only dispersion laws and the structure of the matrix elements. This approach makes it possible to consider a number of problems in a unique way. We discuss the stability of monochromatic waves and the statistical description of a plasma. The connection between decay and modulational instability growth rates and matrix elements is demonstrated. The standard form of the equations enables us to introduce a statistical description in a very simple way. We discuss the usual kinetic wave equations and their generalization for inhomogeneous turbulence and turbulence excited by a coherent pump. We pay special attention to the problem of Langmuir turbulence. The average dynamical equations are deduced in a consistent way and we present a detailed discussion of the limits of this description.