  Weingarten surfaces are those whose principal curvatures satisfy a functional
relation, whose set of solutions is called the curvature diagram or the
W-diagram of the surface. Making use of the notion of geometric linear momentum
of a plane curve, we propose a new approach to the study of rotational
Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing
any type of Weingarten condition on a rotational surface to a first order
differential equation on the momentum of the generatrix curve. In this line, we
provide two new classification results involving a cubic and an hyperbola in
the W-diagram of the surface characterizing, respectively, the non-degenerated
quadric surfaces of revolution and the elasticoids, defined as the rotational
surfaces generated by the rotation of the Euler elastic curves around their
directrix line.
  As another application of our approach, we deal with the problem of
prescribing mean or Gauss curvature on rotational surfaces in terms of
arbitrary continuous functions depending on distance from the surface to the
axis of revolution. As a consequence, we provide simple new proofs of some
classical results concerning rotational surfaces, like Euler's theorem about
minimal ones, Delaunay's theorem on constant mean curvature ones, and Darboux's
theorem about constant Gauss curvature ones.
