Determination of the evolution of a system is studied through the definition of functionals presented here in the case of non linear dynamics. After a short account of the necessary notions for the description of the motion and of the mechanical interactions, the paper is devoted to the Hamiltonian functional of the system. The evolution of discontinuities along moving surfaces generally generates an entropy production which is a linear function of the speed of the moving interface. The thermodynamical force accompanying this motion is an energy release rate, the expression of which is related to the gradient of Hamiltonian of the system. The canonical equations of shock waves and moving discontinuities are established for any continuum. This provides a useful way of establishing constitutive laws for moving interfaces.