  The action principle is frequently used to derive the classical equations of
motion. The action may also be used to associate group elements with curves in
the space-time manifold, similar to the gauge transformations. The action
principle is shown here to be an equivalence relation between the infinitesimal
elements so defined for a collection of closed curves and the identity element.
The action principle is then extended by requiring the equivalence of global
elements with the identity and by considering all curves. The resulting
equation is generalized further to include the non-Abelian gauge fields. The
extended equation has an infinite number, but not all, trajectories as
solutions. The properties of these paths are shown to impart wave-like
properties to the particles in motion. These results provide an insight into
the wave-particle duality and lead to a modified path-integral formalism. The
motion of a particle is formulated within the resulting framework which yields
a generalized Schrodinger equation. This equation is shown to reduce to a set
of equations, one of them being the Klein-Gordon equation.
