Foundation of Tensor Flight Dynamics

D.1 Introduction

In this appendix you find the analytical proofs of the two fundamental elements of tensor flight dynamics: the rotational time operator and the Euler transforma­tion. While Chapter 4 introduced the rotational time derivative in an axiomatic fashion, here I will derive it from general space-time properties. Similarly, in Chapter 4, Euler’s transformation was given a rational proof based on the isotropic property of Euclidean space, whereas tensor calculus will be used here to pro­vide an analytical proof. To follow the exposition you should be skilled in tensor analysis and nimble in manipulating sub – and superscripted variables.

Tensor flight dynamics departs from the methodology of Gibb’s vector mechanics.1 It formulates the equations of motion in a form that is invariant under time-dependent Cartesian coordinate transformations. Although a branch of classi­cal physics, tensor flight dynamics conforms to Einstein’s Principle of Covariance (Ref. 2. p. 107): No coordinate system should be distinguished in the formulation of physical laws. In other words, the physical laws should have the same form, i. e., be covariant (invariant) in all coordinate systems, even in accelerated systems. The covariance principle is the cornerstone of the General Theory of Relativity, whereas in the Special Theory of Relativity the same postulate encompasses only inertial coordinate systems.

In classical dynamics, Newton’s second law is formulated as invariant with re­spect to inertial coordinate systems. If the coordinate transformations are time dependent (rotationally accelerated), it takes on a different form, i. e., the covari­ance principle is violated. The intent of tensor flight dynamics is to impose the covariance principle on the formulation of the equations of motion, in order for them to be invariant also under time-dependent coordinate transformations. Then the physics of flight can be modeled first in tensors, followed by conversion to matrices, and finally coded for computer execution.

Not every subscripted variable is a tensor; though some publications make that claim. An entity is a tensor only, if it maintains its form under all admissible coor­dinate transformations. In classical dynamics, and especially in flight dynamics, coordinate systems are Cartesian with generally time-dependent elements. As soon as ordinary time derivatives are taken, additional terms appear in the equations of motion, which destroy the tensor quality. The special rotational time derivative, however, maintains the tensor property.

The objective of this appendix is to derive the rotational time derivative from general principles. After restricting the space to be Euclidian/Cartesian, the ro­tational time derivative will emerge as it is used in flight dynamics. Proofs will be given of the tensor properties of the rotational time derivative when applied to tensors of rank one and two, as well as to axial vectors. Furthermore, the Euler
transformation, which governs the change of reference frames, also will be given a sound analytical basis.

D.2 Derivation of the Rotational Time Derivative

Any two coordinate systems xk and x‘ embedded in general и-dimensional space may be related by time-independent coordinate transformations (also called sderonomic transformations),

x’ =x'{xk) (i, k = 1, …, и) (D. l)

or by time-dependent (rheonomic) transformations,

x’=x,(xk, t) (i, к = 1, и) (D.2)

The coordinate differential dx’of the barred system is obtained from that of the unbarred system dxA by the local sderonomic transformation,

fix’

dx’ — —Tdxk (D.3)

dxk

or the local rheonomic transformation,

dx‘ = ^-dxk + —dt (D.4)

дхк 3t

The total differential form Sv’ of a contravariant tensor of rank one v’ for sclero- nomic space is

Sv1 =dv‘+f‘jkvjdxk (D.5)

implying Einstein’s summation convention. dS’ is the incremental change of the vector and Г’д D ‘dxA the so-called transplantation contribution from curved space. (Ref. 2, p. 41). Г’-к are called the coefficients of affine connections or simply the connection matrix. The differential form in the rheonomic space includes, according to Wundheiler,3 another term K’-v’ dt that accounts for the changing time at space point xk

Sv’ — dv’ + r’^iVdx^ + A’iv’dt

with Л’ called the rotation matrix.

і

Our aim is to determine the form of the rotation matrix. To proceed, we assume that in the rheonomic space xk there is one location x’where the transformation is independent of time, i. e., the first partial derivatives of space coordinates wrt time are zero. At that location the total differential is given by Eq. (D.5) and the transformation of a vector vk is governed by the transformation

(D.7)

Substituting Eq. (D.7) into Eq. (D.5), we have

but

dxk „ , 3 / Эх’

— ^0 and —

Substituting Eqs. (D.9) and (D.10) into Eq. (D.8) yields

Эх’ 32x’ о v Э2х’ о Эх; дхк о v

Sv = —– du -|—— ——– v^dxy ——- – —u^dt + Г jk——– v^dxY

дха Эх/Эх5′ Эх/Эt ]k дх$ дхУ

Fig. D. l Visualization of partial differentials.

and by rearranging we get

The two terms in parentheses are the transformed connection matrix (Ref. 4, p. 98):

This connection matrix accounts for the effect of the curvature of space. All three terms within the braces of Eq. (D.12) constitute the differential form in rheonomic space. In particular, the contribution of the explicit time dependency is captured in the last term of Eq. (D. 12) by the rotation matrix:

„ _ dxa d2Xh P dxh дх$дt

In summary, the terms in the braces of Eq. (D.12) comprise the differential form 8v’ in rheonomic space:

Now we assume that the space is Euclidean. Because of its flatness, the connec­tion matrix vanishes, but not the rotation matrix:

8 v’ = du’ + A’jVJd t

Introduce new nomenclature for the transformation matrices:

dx‘ _h dxh h = tj =

Their product is the unit tensor, represented by the Kronecker symbol <5′(:

Because the transformation is now only dependent on time, we can write Eq. (D.14) in the following form:

Substituting A’, into Eq. (D.16) and dividing by dt yields the rotational time derivative operator 8/8t applied to v‘:

The rotational time derivative consists of the ordinary time derivative du’/dt and an additional term contributed by the rotation of the coordinate system.

For the remainder of this appendix we limit the discussion to Cartesian coor­dinate systems. In this case the distinction between covariant and contravariant vectors disappears, and we can write Eq. (D.20) in subscripted form only:

8 V( d d

— = ~7~Vj + Uh~7~t jh^j (D.21)

8t d t dt 1 J

Because coordinate transformations in Cartesian space are orthogonal, their inverse equals their transposed; i. e., the inverse th – of tlh, now to,, equals the transposed fh, where the overbar has become superfluous.