  We give a causal version of Eisenhart's geodesic characterization of
classical mechanics. We emphasize the geometric, coordinate independent
properties needed to express Eisenhart's theorem in light of modern studies on
the Bargmann structures (lightlike dimensional reduction, pp-waves). The
construction of the space metric, Coriolis 1-form and scalar potential through
which the theorem is formulated is shown in detail, and in particular it is
proved a one-to-one correspondence between Newtonian frames and Abelian
connections on suitable lightlike principal bundles. The relation of
Eisenhart's theorem in the lightlike case with a Fermat type principle is
pointed out. The operation of lightlike lift is introduced and the existence of
minimizers for the classical action is related to the causal simplicity of
Eisenhart's spacetime.
