  We consider any dynamical system that starts from a given ensemble of
configurations and evolves in time until the system reaches a certain fixed
stopping criterion, with the mean first-passage time the quantity of interest.
We present a general method, Projective Dynamics, which maps the
multi-dimensional dynamics of the system onto an arbitrary discrete set of
states ${\zeta_k}$, subject only to the constraint that the dynamics is
restricted to transitions not further than the neighboring states $\zeta_{k\pm
1}$. We prove that with this imposed condition there exists a master equation
with nearest-neighbor coupling with the same mean first-passage time as the
original dynamical system. We show applications of the method for Brownian
motion of particles in one and two dimensional potential energy landscapes and
the folding process of small bio-polymers. We compare results for the mean
first passage time and the mean folding time obtained with the Projective
Dynamics method with those obtained by a direct measurement, and where possible
with a semi-analytical solution.
