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

Gelfand--Dorfman algebras, derived identities, and the Manin product of operads