New non-perturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to e -6H which are correct to higher order in S and which contain fewer terms than the Campbell-Hausdorff formula. This allows one to'generate path-integral actions which are more accurate for finite

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... e steps in time and which can be exploited to improve the rate of convergence of Monte-Carlo calculations. To show that these methods allow one to include effects which show up in the stationary phase approximation to the path integral, e.g. instantons, solitons, etc., I not only derive the connection between the Hamiltonian and path-integral formalism but the relationship between a specific stationary-phase approximation and the corresponding Hamiltonian calculation. My focus, however, is the direct application of these new identities to the study of non-perturbative Hamiltonian dynamics. I show that these methods are easier to apply and give better results than those based upon the naive t-expansion, block-mean field or real-space renormalization ideas. Comparison of the these older methods with the computational tools introduced in this paper are discussed in the context of simple examples. It is shown that the new methods allow one to extrapolate answers to finite t without the use of Padk approximants.