The L2 or H0 metric on the space of smooth plane regular closed curves induces vanishing geodesic distance on the quotient Imm(S1,R2)/Diff(S1). This is a general phenomenon and holds on all full diffeomorphism groups and spaces Imm(M,N)/Diff(M) for a compact manifold M and a Riemanninan manifold N. Thus we have to consider more complicated Riemannian metrics using lenght or curvature, and we do this is a systematic Hamiltonian way, we derive geodesic equation and split them into horizontal and vertical parts, and compute all conserved quantities via the momentum mappings of several invariance groups (Reparameterizations, motions, and even scalings). The resulting equations are relatives of well known completely integrable systems (Burgers, Camassa Holm, Hunter Saxton).