Ordinarily, the Jacobi formulation of classical mechanics describes a system by giving the path of a “system point” in “configuration space,” the path being a geodesic if a metric is suitably defined. Nothing is said, however, about the motion in time.
It is suggested here that, by adding a degree of freedom in uniform motion to serve as a clock, and identifying the time with the coordinate of that degree of freedom, the configuration space can be extended to a space-time, in which the motion is completely described, both in space and in time, as being along a geodesic. Also, by this representation, the idea of “absolute time” is made unnecessary.
A representation such as this is, of course, normal in relativistic theory, but its use here in nonrelativistic mechanics is believed to have some novelty.