Projective Transformations and the Hamiltonian

In a previous paper I have defined the notions of time, the Hamiltonian and the Lagrangian for a given time measure in terms of a Pfaffian X_idxⁱ of class 2n+l in the 2n+1 variables of the manifold of states and time, representing the action of a dynamical system in an infinitesimal displacement. Adopting the the same point of view, I here derive invariant expressions for the kinetic and the potential energies of the system by restricting the admitted transformations to the projective subgroup of the general analytic transformations.