  The path integral formulation of quantum mechanics constructs the propagator
by evaluating the action S for all classical paths in coordinate space. A
corresponding momentum path integral may also be defined through Fourier
transforms in the endpoints. Although these momentum path integrals are
especially simple for several special cases, no one has, to my knowledge, ever
formally constructed them from all classical paths in momentum space. I show
that this is possible because there exists another classical mechanics based on
an alternate classical action R. Hamilton's Canonical equations result from a
variational principle in both S and R. S uses fixed beginning and ending
spatial points while R uses fixed beginning and ending momentum points. This
alternative action's classical mechanics also includes a Hamilton-Jacobi
equation. I also present some important points concerning the beginning and
ending conditions on the action necessary to apply a Canonical transformation.
These properties explain the failure of the Canonical transformation in the
phase space path integral. It follows that a path integral may be constructed
from classical position paths using S in the coordinate representation or from
classical momentum paths using R in the momentum representation. Several
example calculations are presented that illustrate the simplifications and
practical advantages made possible by this broader view of the path integral.
In particular, the normalized amplitude for a free particle is found without
using the Schrodinger equation, the internal spin degree of freedom is simply
and naturally derived, and the simple harmonic oscillator is calculated.
