Brane variation Dirac style

Dirac's method for variations of a brane embedded in co-dimension one is demonstrated. The variation in the location of the brane invokes a rest frame formulation of the 'sandwiched' brane action. We first demonstrate the necessity of this method by re-deriving Snell's law. Second, we apply the method to a general N-dimensional brane embedded in co-dimension one bulk in the presence of gravity. We re-derive the brane equations: (i) the Israel junction condition, (ii) the energy/momentum conservation on the brane and (iii) a geodetic-type equation for the brane.