The Sobolev embedding theorem arguments were made in demonstrating that 1-continuous derivative is required for smooth flows. The Green's operator generated from the Green's function(scalar case) associated to the differential operator smooths. For proper choice of formula_46 then formula_47 is an RKHS with the operator formula_48. The Green's kernels associated to the differential operator smooths since for controlling enough derivatives in the square-integral sense the kernel formula_49 is continuously differentiable in both variables implying  (I,J)=\inf_{\phi \in \operatorname{Diff}_V: \phi \cdot I = J } d_{\operatorname{Diff}_V}(id,\phi) \ ;  The distance on shapes and forms, formula_51,  For calculating the metric, the geodesics are a dynamical system, the flow of coordinates formula_52 and the control the vector field formula_53 related via formula_54 The Hamiltonian view     reparameterizes the momentum distribution formula_55 in terms of the "Hamiltonian momentum," a Lagrange multiplier formula_56 constraining the Lagrangian velocity formula_57.accordingly: The Pontryagin maximum principle gives the Hamiltonian formula_59 The optimizing vector field formula_60 with dynamics formula_61. Along the geodesic the Hamiltonian is constant: formula_62. The metric distance between coordinate systems connected via the geodesic determined by the induced distance between identity and group element:  For landmarks, formula_64, the Hamiltonian momentum  with Hamiltonian dynamics taking the form  with The metric between landmarks formula_68  The dynamics associated to these geodesics is shown in the accompanying figure. For surfaces, the Hamiltonian momentum is defined across the surface has Hamiltonian  and dynamics  For volumes the Hamiltonian