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The Hamiltonian description of mechanical or field models defined by singularLagrangians plays a central role in physics. A number of methods are known forthis purpose, the most popular of them being the one developed by Dirac. Here,we discuss other approaches to this problem that rely on the direct use of theequations of motion (and the tangency requirements characteristic of the Gotay,Nester, Hinds method), or are formulated in the tangent bundle of theconfiguration space. Owing to its interesting relation with general relativitywe will use a concrete example as a test bed: an extension of the Pontryaginand Husain-Kucha\v{r} actions to four dimensional manifolds with boundary.

Three roads to the geometric constraint formulation of gravitational theories with boundaries