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

Authors
Citation
M. Szydlowski, THE EISENHART GEOMETRY AS AS ALTERNATIVE DESCRIPTION OF DYNAMICS IN TERMS OF GEODESICS, General relativity and gravitation, 30(6), 1998, pp. 887-914
Citations number
53
Categorie Soggetti
Physics
ISSN journal
00017701
Volume
30
Issue
6
Year of publication
1998
Pages
887 - 914
Database
ISI
SICI code
0001-7701(1998)30:6<887:TEGAAA>2.0.ZU;2-K
Abstract
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.