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We give a causal version of Eisenhart's geodesic characterization ofclassical mechanics. We emphasize the geometric, coordinate independentproperties needed to express Eisenhart's theorem in light of modern studies onthe Bargmann structures (lightlike dimensional reduction, pp-waves). Theconstruction of the space metric, Coriolis 1-form and scalar potential throughwhich the theorem is formulated is shown in detail, and in particular it isproved a one-to-one correspondence between Newtonian frames and Abelianconnections on suitable lightlike principal bundles. The relation ofEisenhart's theorem in the lightlike case with a Fermat type principle ispointed out. The operation of lightlike lift is introduced and the existence ofminimizers for the classical action is related to the causal simplicity ofEisenhart's spacetime.Comment: Latex2e, 29 pages, 2 figures, uses psfrag package. v2: a problem with  theorem 5.5, which was erroneously stated twice, and some misprints have been  fixed, v3: change of definition of function I in theorem 5.

Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime