Technical report | On the Use of Vectors, Reference Frames, and Coordinate Systems in Aerospace Analysis

Abstract

This report describes the core foundational concepts of aerospace modelling, an understanding of which is necessary for the analysis of complex environments that hold many entities, each of which might employ a separate reference frame and coordinate system. I begin by defining vectors, frames, and coordinate systems, and then discuss the quantities that allow Newton's laws to be applied to a complex scenario. In particular, I explain the crucial distinction between the ‘coordinates of the time-derivative of a vector" and the \time-derivative of the coordinates of a vector’. I finish by drawing a parallel between this aerospace language and the notation found in seemingly unrelated areas such as relativity theory and fluid dynamics, and make some comments on various supposedly different derivatives as found in the literature.

Executive Summary

Classical concepts of kinematics are by now well established. But the many differences of opinion on the subject easily found on various internet physics and maths discussion sites indicate that despite being well established, these concepts are not necessarily well understood. I believe that this confusion stems from standard textbook presentations on the subject, whose applicability is limited to only very simple scenarios. In the complex environments encountered in aerospace, a far more advanced understanding of the relevant concepts and their notation is pivotal to the success of any analysis.

This report discusses the more advanced (but still standard) definitions of vectors, reference frames, and coordinate systems that allow their use to model the most complex aerospace scenarios. I begin by defining vectors, highlighting the crucial distinction between a proper vector (an arrow) and a coordinate vector (an array of numbers). Next I define a reference frame as a quasi-physical scaffold relative to which all motion is defined. I then define a coordinate system essentially as a set of rulers attached to a frame, but with the proviso that one's choice of frame need not be tied to one's choice of coordinates; that is, we are free to quantify the events in our chosen frame by using the coordinates natural to another frame.

Having defined the basic concepts, I discuss the quantities that allow Newton's laws to be applied to a complex system. In particular, I explain the important difference between the ‘coordinates of the time-derivative of a vector’ and the ‘time-derivative of the coordinates of a vector’, which is central to aerospace calculations. This difference is not widely appreciated in the field, nor indeed in physics more generally, where it is generally taught only in advanced courses in relativity — but where its meaning is easily lost in a forest of notation. And yet this difference is a basic part of vector analysis that could easily be taught at a first-year university level.

To unify the discussions of this report with the bigger picture, I finish by drawing a parallel between this aerospace language and the tensor notation common in relativity theory. One theme of this report is that some apparently different types of derivative found in the literature are, in essence, identical: although often viewed as ‘new’ or special, they are in fact nothing more than standard derivatives written in a way that is meant to aid practitioners in the various fields that use them.