REDUCED HAMILTONIANS IN GENERAL

In this first of a pair of papers we resurrect a standard construction of analytical mechanics dating from the last century. The technique a llows one to pass from any dynamical system whose first order evolutio n equations are known, and whose bracket algebra is not degenerate, to a system of canonical variables and a non-zero Hamiltonian that gener ates their evolution. The construction agrees with the usual results f or gauge theories and can be applied as well to gravity, even when the spatial manifold is closed. Although our results are classical they c an be formally quantized to give the naive functional formalism. This is not only an effective starting point for calculations, it also seem s to provide a formulation of the quantum theory which is non-perturba tive, at least in principle.