Absolute Time Derivatives

TL;DR

This paper presents a four-dimensional framework for objective time derivatives in continuum mechanics, showing how Lie derivatives naturally describe these derivatives and analyzing their forms for various tensor types from a rotating observer's perspective.

Contribution

It introduces a four-dimensional approach to objective time derivatives, clarifies their relation to Lie derivatives, and derives explicit forms for different tensor quantities in a rotating frame.

Findings

Lie derivatives represent objective time derivatives naturally.

Explicit forms of derivatives for scalars, vectors, and tensors are derived.

The relation between substantial, material, and objective derivatives is clarified.

Abstract

A four dimensional treatment of nonrelativistic space-time gives a natural frame to deal with objective time derivatives. In this framework some well known objective time derivatives of continuum mechanics appear as Lie-derivatives. Their coordinatized forms depends on the tensorial properties of the relevant physical quantities. We calculate the particular forms of objective time derivatives for scalars, vectors, covectors and different second order tensors from the point of view of a rotating observer. The relation of substantial, material and objective time derivatives is treated.