DYNAMICAL TRAJECTORIES OF SIMPLE MECHANICAL SYSTEMS AS GEODESICS IN SPACE WITH AN EXTRA DIMENSION

We show the advantages of representing the dynamics of simple mechanic al systems described by a natural Lagrangian, in terms of geodesics of a Riemannian (or pseudo-Riemannian) space with an additional dimensio n. We demonstrate how general trajectories of simple mechanical system s can be put into one-to-one correspondence with the geodesics of a su itable manifold. Two different ways in which the geometry of the confi guration space can be obtained from a higher dimensional model are pre sented and compared: (1) by a straightforward projection, and (2) as a space geometry of a quotient space obtained by the action of the time like Killing vector generating a. stationary symmetry of a background space geometry with an additional dimension. The second model is more informative and coincides with the so-called optical model of the line -of-sight geometry. On the base of this model we study the behavior of nearby geodesics to detect their sensitive dependence on initial cond itions-the key ingredient of deterministic chaos. The advantage of suc h a formulation is its invariant character.