The Eisenhart Geometry as an Alternative Description of Dynamics in Terms of Geodesics

Details The Eisenhart Geometry as an Alternative Description of Dynamics in Terms of Geodesics The Eisenhart Geometry as an Alternative Description of Dynamics in Terms of Geodesics

Jun 1998

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1998gregr..30..887s&link_type=abstract

General Relativity and Gravitation, Volume 30, Issue 6, pp.887-914

Computer Science

6

Particle Trajectories, Deterministic Chaos

Scientific paper

We show the advantages of representing the dynamics of simple mechanical systems, described by a natural Lagrangian, in terms of geodesics of a Riemannian (or pseudo-Riemannian) space with an additional dimension. We demonstrate how trajectories of simple mechanical systems can be put into one-to-one correspondence with the geodesics of a suitable manifold. Two different ways in which geometry of the configuration space can be obtained from a higher dimensional model are presented and compared: First, by a straightforward projection, and second, as a space 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 informative and coincides with the so-called optical model of the line of sight geometry. On the base of this model we study the behaviour of nearby geodesics to detect their sensitive dependence on initial conditions—the key ingredient of deterministic chaos. The advantage of such a formulation is its invariant character.

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