非仿射几何化会导致非物理的不稳定性
Non-affine geometrization can lead to nonphysical instabilities
摘要：
动力学的几何化包括在具有适当定义的度量的配置空间上用测地线表示轨迹。以前，努力表明，动态稳定性的分析也可以在几何框架内进行，通过测量一束测地线的加宽率。两个已知的形式主义是通过雅可比和Escherart度量。我们发现，这种几何分析措施的实际稳定性时，任何测地线的长度是成比例的相应的时间间隔。我们证明了雅可比度量并不总是一个适当的参数化，它预测的谐振子系统的混沌行为。此外，我们表明，通过明确的计算，动力学和几何传播之间的对应关系是不明确的雅可比度量。我们发现，Escherart动力学对应于实际的切线动力学，因此是一个适当的几何化方案。
英文摘要：
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.
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