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Reduced Hamiltonians

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.