  Classical mechanics for individual physical systems and quantum mechanics of
non-relativistic particles are shown to be exceptional cases of a generalized
dynamics described in terms of maps between two manifolds, the source being
configuration space. The target space is argued to be 2-dimensional and flat,
and their orthogonal directions are physically interpreted. All terms in the
map equation have a geometrical meaning in the target space, and the pull-back
of its rotational Killing one-form allows the construction of a velocity field
in configuration space. Identification of this velocity field with tangent
vectors in the source space leads to the dynamical law of motion. For a
specific choice of an arbitrary scalar function present in the map equation,
and using Cartesian coordinates in the target space, the map equation becomes
linear and can be reduced to the Schr\"odinger equation. We link the
bi-dimensionality of the target space with the essential non-locality of
quantum mechanics. Many extensions of the framework here presented are
immediate, with deep consequences yet to be explored.
