  We resurrect a standard construction of analytical mechanics dating from the
last century. The technique allows one to pass from any dynamical system whose
first order evolution equations are known, and whose bracket algebra is not
degenerate, to a system of canonical variables and a non-zero Hamiltonian that
generates their evolution. We advocate using this method to infer a canonical
formalism, as a prelude to quantization, for systems in which the naive
Hamiltonian is constrained to vanish. The construction agrees with the usual
results for gauge theories and can be applied as well to gravity, {\it even
when the spatial manifold is closed.} As an example, we construct such a
reduced Hamiltonian in perturbation theory around a flat background on the
manifold $T^3 \times R$. The resulting Hamiltonian is positive semidefinite and
agrees with the A.D.M. energy in the limit that deviations from flat space
remain localized as the toroidal radii become infinite.
