  A technique for reducing the number of integrals in a Monte Carlo calculation
is introduced. For integrations relying on classical or mean-field trajectories
with local weighting functions, it is possible to integrate analytically at
least half of the integration variables prior to setting up the particular
Monte Carlo calculation of interest, in some cases more. Proper accounting of
invariant phase space structures shows the system's dynamics is reducible into
composite stable and unstable degrees of freedom. Stable degrees of freedom
behave locally in the reduced dimensional phase space exactly as an analogous
integrable system would. Classification of the unstable degrees of freedom is
dependent upon the degree of chaos present in the dynamics. The techniques for
deriving the requisite canonical coordinate transformations are developed and
shown to block diagonalize the stability matrix into irreducible parts. In
doing so, it is demonstrated how to reduce the amount of sampling directions
necessary in a Monte Carlo simulation. The technique is illustrated by
calculating return probabilities and expectation values for different dynamical
regimes of a two-degree-of-freedom coupled quartic oscillator within a
classical Wigner method framework.
