  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.
