The group of diffeomorphisms of a compact manifold endowed with the L 2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is the natural extension to the classical L 2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the H div right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa-Holm equation in one dimension. This point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. The main application consists in writing the Camassa-Holm equation on S 1 as a particular case of the incompressible Euler equation on a group of homeomorphisms of R 2 that preserve a radial density which has a singularity at 0. An other application consists in proving that smooth solutions of the Euler-Arnold equation for the H div right-invariant metric are length minimizing geodesics for sufficiently short times.