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

Citation
M. Szydlowski et al., DYNAMICAL TRAJECTORIES OF SIMPLE MECHANICAL SYSTEMS AS GEODESICS IN SPACE WITH AN EXTRA DIMENSION, International journal of theoretical physics, 37(5), 1998, pp. 1569-1585
Citations number
28
Categorie Soggetti
Physics
ISSN journal
00207748
Volume
37
Issue
5
Year of publication
1998
Pages
1569 - 1585
Database
ISI
SICI code
0020-7748(1998)37:5<1569:DTOSMS>2.0.ZU;2-A
Abstract
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.