On the minimal set of conservation laws and the Hamiltonian structure of the Whitham equations

We consider the questions connected with the Hamiltonian properties of the Whitham equations in case of several spatial dimensions. An essential point of our approach here is a connection of the Hamiltonian structure of the Whitham system with the finite-dimensional Poisson bracket defined on the space of periodic or quasi-periodic solutions. From our point of view, this approach gives a possibility to construct the Hamiltonian structure of the Whitham equations under minimal requirements on the properties of the initial system. The Poisson bracket for the Whitham system can be considered here as a deformation of the finite-dimensional bracket with the aid of the Dubrovin-Novikov procedure of bracket averaging. At the end, we consider the examples where the constructions of the paper play an essential role for the construction of the Poisson bracket for the Whitham system.