  Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two
bilinear operations satisfying a series of identities that express Hamiltonian
property of an operator in the formal calculus of variations. The paper is
devoted to the study of GD-algebras related with differential Poisson algebras.
As a byproduct, we obtain a general description of identities that hold for
operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative)
differential algebra with a derivation~$d$.
