Poincare-cartan integral invariant and canonical transformations for singular Lagrangians:An addendum

In a previous paper) we discussed in the framework of the Poincare-Cartan integral invariant, a method for performing the canonical formalism for constrained systems. The basic idea consists of considering a canonical transformation which brings the constraints into a subset of the canonical variables. Thus the physical variables can be easily obtained by means of a reduction of the phase space. Our method is different from the path-integral approach of Faddeev (see also Ref. 3), which use in addition a set of gaugefixing conditions, one for each first-class constraint. Two applications of our procedure concerning action-at-a-distance relativistic models have been recently studied. In this note we extend the method by considering a time-dependent general canonical transformation, such that all the constraints acquire an explicit time dependence. Let us consider a dynamical system described in terms of2n degrees offreedom in the phase space qs'Ps (s = I, ... ,n) and constrained to the hypersurface S defined by