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

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