Propagation of surfaces of weak discontinuity as behaviour of broken extremals in classical field theory

A variational method is shown to derive the equations describing the propagation of surfaces of weak discontinuity, i.e. of those surfaces across which at least one of the derivatives of the field variables suffer a jump. From the point of view of the calculus of variations this is the case of the broken extremals. The equations are derived from that variational principle which accounts for the non-broken extremals. The variational principle must be formulated at least in the frame of the special relativity.