= Hamiltonian formulation of general relativity
{c}
Using the <lapse function> $N$, <shift vector> $N^i$ and spatial metric $h_{ij}$, the <Legendre transform in mechanics> of the <Einstein-Hilbert action> gives, in units $16\pi G=1$,
$$
H_{\rm bulk}=\int d^3x\sqrt h\,(N\mathcal H+N^i\mathcal H_i),\qquad\mathcal H=\frac{\pi^{ij}\pi_{ij}-\pi^2/2}{h}-{}^{(3)}R,\qquad\mathcal H_i=-2D_j\left(\frac{\pi^j{}_i}{\sqrt h}\right).
$$
Here $\pi=h_{ij}\pi^{ij}$ and $\pi^{ij}$ is the <canonical momentum of the spatial metric>. Varying lapse and shift imposes the <Hamiltonian constraint> and <momentum constraint>. An asymptotic time translation requires an <Arnowitt-Deser-Misner energy> boundary term. The bulk constraints generate gauge evolution; the nonzero asymptotic energy comes from the boundary.
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