Wave interactions by Hamiltonian methods

This thesis is devoted to the description of fluids and gases from a Hamiltonian point of view. The method we apply is a development of the theory invented by Hamilton (1805-1865). The original formulation is restricted to a certain family of dynamical variables; the canonical variables. In fluid dynamics, however, an extension to a noncanonical formulation is necessary, at least if we want to benefit from the Hamiltonian machinery and simultaneously want to use variables with a simple physical interpretation. The noncanonical Hamiltonian structure of the fluid equations can be utilized for several applications. We apply it to nonlinear wave interaction problems for fluid descriptions like the vorticity equation, the shallow-water equation, a family of Hasegawa-Mima-like equations and the equations describing dusty magnetoplasmas. The Hamiltonian structure is used to simplify the calculation of the strength of the coupling between the waves. Another important application concerns the question of finding fluid states that are stable against small perturbations. The Hamiltonian formulation is in this case an effective tool, and the derivation of explicit stability criteria is simplified. The first chapter of this thesis gives a brief and informal introduction to fluid dynamics. In chapter two we present, in a more formal way, some important fluid equations. Chapter three contains a discussion of variational principles and offers a first glance of Hamiltonian theory. The generalisation of this theory is the main subject of the first sections of chapter four, and we finish this chapter by presenting some applications of the general Hamiltonian theory. These applications which involve several different fluid systems are presented in the six papers included in this thesis. In the main, they concern the explicit calculation of symmetrical coupling coefficients and the derivation of sufficient conditions for stability. Some of the results, however, deals with the influence of boundaries in the vorticity equation and the construction of variational principles for the linearised system.digitalisering@um