Based on the insight gained by many authors over the years on the structure of the Einstein-Hilbert, Gauss-Bonnet and Lovelock gravity Lagrangians, we show how to derive -- in an elementary fashion -- their first-order, generalized "ADM" Lagrangian and associated Hamiltonian. To do so, we start from the Lovelock Lagrangian supplemented with the Myers boundary term, which guarantees a Dirichlet variational principle with a surface term of the form $\pi^{ ij}\delta h_{ ij}$, where $\pi^{ ij}$ is the canonical momentum conjugate to the boundary metric $h_{ ij}$. Then, the first-order Lagrangian density is obtained either by integration of $\pi^{ ij}$ over the metric derivative $\partial_wh_{ ij}$ normal to the boundary, or by rewriting the Myers term as a bulk term.
Categories: gr-qc hep-th