THE EISENHART GEOMETRY AS AS ALTERNATIVE DESCRIPTION OF DYNAMICS IN TERMS OF GEODESICS

We show the advantages of representing the dynamics of simple mechanic al systems, described by a natural Lagrangian, in terms of geodesics o f a Riemannian (or pseudo-Riemannian) space with an additional dimensi on. We demonstrate how trajectories of simple mechanical systems can b e put into one-to-one correspondence with the geodesics of a suitable manifold. Two different ways in which geometry of the configuration sp ace can be obtained from a higher dimensional model are presented and compared: First, by a straightforward projection, and second, as a spa ce geometry of a quotient space obtained by the action of the timelike Killing vector generating a stationary symmetry of a background space geometry with an additional dimension. The second model is more infor mative and coincides with the so-called optical model of the line of s ight geometry. On the base of this model we study the behaviour of nea rby geodesics to detect their sensitive dependence on initial conditio ns-the key ingredient of deterministic chaos. The advantage of such a formulation is its invariant character.