Non-affine geometrization can lead to nonphysical instabilities

Abstract

Geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical- and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme.

Featured on the Cover of the Journal. 9 pages, 6 figures: http://iopscience.iop.org/1751-8121/48/7/075101

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• Geometry of dynamics, Lyapunov exponents and phase transitions
• Relativistic chaos is coordinate invariant
• Lyapunov exponents from geodesic spread in configuration space
• Relativistic invariance of Lyapunov exponents in bounded and unbounded systems
• (Non)Invariance of dynamical quantities for orbit equivalent flows
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Cited by in corpus (3)

• Finslerian geometrization of quantum mechanics in the hydrodynamical representation
• Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric
• Hamiltonian chaos and differential geometry of configuration space-time