On Hamiltonian operators in differential algebras

Poisson brackets are widely used to write finite dimensional evolution systems in the Hamiltonian form. Starting with the famous KdV equation this method is applied to infinite dimensional evolution systems, described by partial differential equations (PDE). To use this technique for PDE in full one needs an efficient machinery for construction Poisson brackets. Such program was initiated by Olver in his book “Applications of Lie Groups to Differential Equations”. Here I present a development of his studies. Namely, in a general algebraic frames the system of equations on coefficients of a linear operator in total partial derivatives to be Hamiltonian, i.e., to satisfy the skew-symmetry and Jacobi conditions is derived. These equations are solved in some simplest cases, including the evolution with the zero-divergence constraints.