  Using a simple geometrical construction based upon the linear action of the
Heisenberg--Weyl group we deduce a new nonlinear Schr\"{o}dinger equation that
provides an exact dynamic and energetic model of any classical system
whatsoever, be it integrable, nonintegrable or chaotic. Within our model
classical phase space points are represented by equivalence classes of
wavefunctions that have identical position and momentum expectation values.
Transport of these equivalence classes without dispersion leads to a system of
wavefunction dynamics such that the expectation values track classical
trajectories {\em precisely\/} for arbitrarily long times. Interestingly, the
value of $\hbar$ proves immaterial for the purpose of constructing this
alternative representation of classical point mechanics. The new feature which
$\hbar$ does mediate concerns a simple embedding of the quantum geometric phase
within classical mechanics. We discuss problems of physical interpretation and
discover a simple route to recover the ordinary linear Schr\"{o}dinger
equation.
