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Covariant Time Derivatives for Dynamical Systems

Jean-Luc Thiffeault
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Abstract

We present a unified derivation of covariant time derivatives, which transform as tensors under a time-dependent coordinate change. Such derivatives are essential for formulating physical laws in a frame-independent manner. Three specific derivatives are described: convective, corotational, and directional. The covariance is made explicit by working in arbitrary time-dependent coordinates, instead of restricting to Eulerian (fixed) or Lagrangian (material) coordinates. The commutator of covariant time and space derivatives is interpreted in terms of a time-curvature that shares many properties of the Riemann curvature tensor, and reflects nontrivial time-dependence of the metric.

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Jean-Luc Thiffeault. 2001-07-23. Covariant Time Derivatives for Dynamical Systems. https://doi.org/10.1088/0305-4470%2F34%2F29%2F309

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