  The description of the $T\bar{T}$ deformation in terms of two-dimensional
gravity is analyzed from the Hamiltonian point of view, in a manner analogous
to the ADM description of general relativity. We find that the Hamiltonian
constraints of the theory imply relations between target-space momentum at
finite volume which are equivalent to the $T\bar{T}$ finite volume flow
equations. This fully-quantum $T\bar{T}$ result emerges already at the
classical level within the gravitational theory. We exemplify the analysis for
the case when the undeformed sector is a collection of $D-2$ free massless
scalars, where it is shown that -- somewhat non-trivially -- the target-space
two-dimensional Poincar\'e symmetry is extended to $D$ dimensions. The
connection between canonical quantization of this constrained Hamiltonian
system and previous path integral quantizations is also discussed. We extend
our analysis to the ``gravitational'' description of $J\bar{T}$-type
deformations, where it is found that the flow equations obtained involve
deformations that twist the spatial boundary conditions.
