Conservation of Hamiltonian structures in Whitham's averaging method

In this paper we consider Whitham's averaging method for systems with a local field-theoretic Hamiltonian structure. We prove that such a Hamiltonian structure is conserved in the process. The study is based on the procedure (suggested by Dubrovin and Novikov) of averaging the local Poisson bracket, for which we establish the necessary properties of conservation of the Jacobi identity and invariance.