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Abstract: Geometrization of dynamics consists of representing trajectories by geodesicson a configuration space with a suitably defined metric. Previously, effortswere made to show that the analysis of dynamical stability can also be carriedout within geometrical frameworks, by measuring the broadening rate of a bundleof geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. Wefind that this geometrical analysis measures the actual stability when thelength of any geodesic is proportional to the corresponding time interval. Weprove that the Jacobi metric is not always an appropriate parametrization byshowing that it predicts chaotic behavior for a system of harmonic oscillators.Furthermore, we show, by explicit calculation, that the correspondence betweendynamical- and geometrical-spread is ill-defined for the Jacobi metric. We findthat the Eisenhart dynamics corresponds to the actual tangent dynamics and istherefore an appropriate geometrization scheme.



Author: Eduardo Cuervo-Reyes, Ramis Movassagh

Source: https://arxiv.org/



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