Dynamical Trajectories of Simple Mechanical Systems as Geodesics in Space with an Extra Dimension

Abstract

We show the advantages of representing thedynamics of simple mechanical systems described by anatural Lagrangian, in terms of geodesics of aRiemannian (or pseudo-Riemannian) space with anadditional dimension. We demonstrate how generaltrajectories of simple mechanical systems can be putinto one-toone correspondence with the geodesics of asuitable manifold. Two different ways in which thegeometry of the configuration space can be obtained froma higher dimensional model are presented and compared:(1) by a straightforward projection, and (2) as a spacegeometry of a quotient space obtained by the action of the timelike Killing vectorgenerating a stationary symmetry of a background spacegeometry with an additional dimension. The second modelis more informative and coincides with the so-called optical model of the line-of-sight geometry. Onthe base of this model we study the behavior of nearbygeodesics to detect their sensitive dependence oninitial conditions — the key ingredient ofdeterministic chaos. The advantage of such a formulation isits invariant character.