On wave action and phase in the non-canonical Hamiltonian formulation

The long time-evolution of disturbances to slowly varying solutions of part ial differential equations is subject to the adiabatic invariance of the wa ve action. Generally, this approximate conservation law is obtained under t he assumption that the partial differential equations are derived from a va riational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non-canonical (Poisson) Hamiltonian structure. The linear evolution of dis turbances in the form of slowly varying wavetrains is studied using a WKB e xpansion. The properties of the original Hamiltonian system strongly constr ain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approx imate) invariance and the (exact) conservation laws of pseud-energy and pse udomomentum that exist when the basic solution is exactly time and space in dependent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.