  We present a systematic method for dealing with time dependent quantum
dynamics, based on the quantum brachistochrone and matrix mechanics. We derive
the explicit time dependence of the Hamiltonian operator for a number of
constrained finite systems from this formalism. Once this has been achieved we
go on to calculate the wavevector as a function of time, in order to
demonstrate the use of matrix methods with respect to several concrete
examples. Interesting results are derived for elliptic curves and periodic
orbits on higher dimensional non-commutative geometries.
