Linear constraints from generally covariant systems with quadratic constraints

How to make compatible both boundary and gauge conditions for generally covariant theories using the gauge symmetry generated by first class constraints is studied. This approach employs finite gauge transformations in contrast with previous works which use infinitesimal ones. Two kinds of variational principles are taken into account; the first one features non-gauge-invariant actions whereas the second includes fully gauge-invariant actions. Furthermore, it is shown that it is possible to rewrite fully gauge-invariant actions featuring first class constraints quadratic in the momenta into first class constraints linear in the momenta (and homogeneous in some cases) due to the full gauge invariance of their actions. This shows that the gauge symmetry present in generally covariant theories having first class constraints quadratic in the momenta is not of a different kind with respect to the one of theories with first class constraints linear in the momenta if fully gauge-invariant actions are taken into account for the former theories. These ideas are implemented for the parametrized relativistic free particle, parametrized harmonic oscillator, and the $\mathrm{SL}(2,R)\mathrm{}$ model.