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AMathematics ofFinite Rotations

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Appendix A: MATHEMATICS OF FINITE ROTATIONS A2

TABLE OF CONTENTS

Page

A.1. Introduction A3A.2. Plane vs. Spatial Rotations A3

A.3. Spinors A4

A.3.1. Spin Matrix and Axial Vector . . . . . . . . . . . . A4

A.3.2. Normalizations . . . . . . . . . . . . . . . . A5

A.3.3. Spectral Properties . . . . . . . . . . . . . . . A6

A.4. From Spinors To Rotators A7

A.4.1. The Algebraic Approach . . . . . . . . . . . . . A7

A.4.2. The Geometric Approach . . . . . . . . . . . . . A8

A.4.3. The Cayley Transform . . . . . . . . . . . . . . A9

A.4.4. Exponential Map . . . . . . . . . . . . . . . . A10A.5. From Rotators to Spinors A10

A.6. Rotator Derivatives A11

A.7. Spinor and Rotator Transformations A12

A.8. Axial Vector Jacobian A12

A.9. Spinor and Rotator Differentiation A13

A.9.1. Angular Velocities . . . . . . . . . . . . . . . A13

A.9.2. Angular Accelerations . . . . . . . . . . . . . . A14

A.9.3. Variations . . . . . . . . . . . . . . . . . . A14

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A3 A.2 PLANE VS. SPATIAL ROTATIONS

A.1. Introduction

This Appendix provides a compendium of formulas and results concerning the mathematical treat-

ment offinite rotations in 3D space. Emphasis is placed on matrix representations of use in the

corotational formulation offinite elements.

The material is taken from a survey paper: C. A. Felippa and B. Haugen, A unified formulationof small-strain corotational finite elements - I. Theory, Comp. Meths. Appl. Mech. Engrg., 194,

22852336, 2005.

A.2. Plane vs. Spatial Rotations

Plane rotations make up an easy topic. A rotation in e.g., the x y plane, is defined by just a scalar:

the rotation angle about z. Plane rotations commute: 1 + 2 = 2 + 1, because the s arenumbers.

The study of spatial rotations ismore difficult. The subject is dominatedby thefundamental theorem

of Euler:

The general displacement of a rigid body with one point fixed is a rotation

about some axis which passes through that point.

Consequently 3D rotations have both magnitude: the angle of rotation, and direction: the axis of

rotation. These are nominally the same two attributes that categorize vectors. Not coincidentally,

rotations are often depicted as vectors but with a double arrow.

Finite spatial rotations, however, do not obey the laws of vector calculus, although infinitesimal

rotations do. Most striking is failure of commutativity: switching two successive rotations does

not yield the same answer unless the axis of rotation is kept fixed. Within the framework of matrix

algebra, finite rotations can be represented as either 3 3 real orthogonal matrices R called rotatorsor as real 3 3 skew-symmetric matrices called spinors.The spinor representation is important in physical modeling and theoretical derivations, because

the matrix entries are closely related to the two foregoing attributes. The rotator representation is

important in numerical computations, as well as being naturally related to polar factorizations of a

transformation matrix. The two representations are related by various transformations illustrated

in Figure A.1. Of these the Cayley transform (1857) is the oldest although not the most important

one.

Axial-vector or

pseudo-vector Rotator

RSkew(R)axial( )

Spinor

Rot( )

Spin( )

Not unique: adjustable

by a scale factor

Unique

Figure A.1. Representations offinite space rotations and mapping operations.

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x3

x2x1

x

r

P(x)

P(x)

C

C

O

Q(x,)

Q(x,)

/2

/2

r

2rsin/2

2rcos/2

(a)

(b)

x

Rotation axis definedby axial vectorCan be normalizedin various ways

Figure A.2. Attributes of rotation in 3D.

Now a 3

3 skew-symmetric matrix is defined by three scalar parameters. These three numbers

can be arranged as components of an axial vector. Although looks like a vector, it does

not obey certain properties of classical vectors such as the composition rule. Therefore the term

pseudo-vectoris sometimes used for .

This overview is intended to stress that finite 3D rotations can appear in various mathematical

representations, as depicted in Figure A.1. The exposition that follows expands on this topic, and

studies the connector links shown in Figure A.1.

A.3. Spinors

Figure A.2(a) depicts a 3D rotation in space (x1,x2,x3) by an angle about an axis of rotation .For convenience the origin of coordinates O is placed on

. The rotation axis is defined by three

directors: 1, 2, 3, at least one of which must be nonzero. These numbers may be scaled by an

nonzero factor through which the vector may be normalized in various ways as discussed later.

The positive sense of obeys the RHS screw rule.

The rotation takes an arbitrary point P(x), located by its position vector x, into Q(x, ), located by

its position vector x . The center of rotation Cis defined by projecting P on the rotation axis. The

plane of rotation CPQ is normal to that axis at C. The radius of rotation is vector r of magnitude r

from Cto P . As illustrated in Figure A.2(b), the distance between P and Q is 2rsin 12

.

A.3.1. Spin Matrix and Axial Vector

Giventhethreedirectors 1, 2,and 3 oftheaxis, wecanassociate withit a 3

3 skew-symmetric

matrix, called a spin tensoror spin matrix, by the rule

= Spin() =

0 3 23 0 1

2 3 0

= T. (A.1)

Premultiplication of a vector v by is equivalent to the cross product of and v:

= v = v. (A.2)

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A5 A.3 SPINORS

In particular: = 0, as may be directly verified.The converse operation to (A.1) extracts the 3-vector , called pseudovector, or axial vector, from

a given spin tensor:

=axial()

= 12

3 . (A.3)

The length of this vector is denoted by :

= || = +

21 + 22 + 23. (A.4)As general notational rule, we will use corresponding upper and lower case symbols for the spin

matrix and its axial vector, respectively. For example, N and n, and , B and b.

A.3.2. Normalizations

As noted, and can be multiplied by a nonzero scalar factor to obtain various normalizations.

In general has the form g()/, where g(.) is a function of the rotation angle . The purposeof normalizations is to simplify the connections Rot and Skew to the rotator, to avoid singularities

for special angles, and to connect the components 1, 2 and 3 closely to the rotation amplitude.

This section review some normalizations that have practical or historical importance.

Taking = 1/ we obtain the unit axial-vector and unit spinor, which are denoted by n and N,respectively:

n =

n1n2n3

=

1/

2/

3/

=

, N = Spin(n) =

=

0 n3 n2n3 0 n1

n2 n3 0

. (A.5)

Taking =

tan 1

2 / is equivalent to multiplying the ni by tan

1

2 . We thus obtain the Rodrigues

parameters bi = ni tan 12 , i = 1, 2, 3. These are collected in the Rodrigues axial-vector b withassociated spinor B:

b = tan 12

n = tan12

, B = Spin(b) = tan 1

2 N = tan

12

. (A.6)

This representation permits an elegant formulation of the rotator via the Cayley transform studied

later. It collapses, however, as nears 180 since tan 12

as 180. One way tocircumvent the singularity is through the use of the four Euler-Rodrigues parameters, also called

quaternion coefficients:

p0 = cos 12 , pi = ni sin 12 = i / sin 12 , i = 1, 2, 3. (A.7)under the constraint p20 +p21 +p22 +p23 = 1. This set is often used in multibody dynamics, roboticsand control. It comes at the cost of carrying along an extra parameter and an additional constraint.

A related singularity-free normalization, introduced by Fraeijs de Veubeke [243], takes =sin 1

2 /. It is equivalent to using only the last three parameters of (A.7):

p = sin 12

n = (sin 12

/) , P = Spin(p) = sin 12

N = (sin 12

/) . (A.8)

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Finally, an important normalization that preserves three parameters while avoiding singularities is

that associated with the exponential map. Introduce a rotation vector defined as

= n = (/), = Spin() = N = (/). (A.9)

For this normalization the angle is the length of the rotation vector: = |

| = 21 + 22 + 23 .The selection of the sign is a matter of convention.

A.3.3. Spectral Properties

Study of the spinor eigensystem vi = i vi is of interest for various developments. Begin byforming the characteristic equation

det( I) = 3 2 = 0, (A.10)where I denotes the identity matrix of order 3. It follows that the eigenvalues of are 1 = 0,2,3 = i . Consequently is singular with rank 2 if = 0 whereas if = 0, is null.

The eigenvalues are collected in thediagonal matrix = diag(0, i, i) and the correspondingright eigenvectors vi in columns ofV = [ v1 v2 v3 ], so that V = V. A cyclic-symmetric

expression ofV, obtained through Mathematica, is

V =

1 1s 2 + i(2 3) 1s 2 i(2 3)2 2s 2 + i(3 1) 2s 2 i(3 1)3 3s 2 + i(1 2) 3s 2 i(1 2)

(A.11)

where s = 1 + 2 + 3. Its inverse is

V1

=1

2

1 1222 + 23

1s 2 i(2 3) 12

22 + 231s 2 + i(2 3)

2 12

23+

212s 2 i(3 1)

12

23+

212s 2 + i(3 1)

3 1221 + 22

3s 2 i(1 2) 12

21 + 223s 2 + i(1 2)

(A.12)

The real and imaginary part of the eigenvectors v2 and v3 are orthogonal. This is a general property

of skew-symmetric matrices; cf. Bellman [67, p. 64].

Because the eigenvalues of are distinct if = 0, an arbitrary matrix function F() can beexplicitly obtained [314,309] as

F() = Vf(0) 0 0

0 f(i) 0

0 0 f(i)V1, (A.13)

in which f(.) is the scalar version ofF(.). One important application of (A.13) is the matrix

exponential: f(.) e(.).The square of, computed through direct multiplication, is

2 =

22 + 23 12 1312 23 + 21 2313 23 21 + 22

= T 2I = 2(nnT I). (A.14)

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A7 A.4 FROM SPINORS TO ROTATORS

This is a symmetric matrix of trace 22 whose eigenvalues are 0, 2 and 2.By theCayley-Hamiltontheorem [67,309,314,636] satisfies itsown characteristic equation (A.10)

3 = 2, 4 = 22, . . . and generally n = 2n2, n 3. (A.15)

Hence ifn = 3, 5, . . . the odd powersn are skew-symmetric with distinct purely imaginary eigen-values, whereas ifn = 4, 6 . . ., the even powersn are symmetric with repeated real eigenvalues.The eigenvalues ofI + and I , are (1, 1 i) and (1, 1 i), respectively. Hencethose two matrices are guaranteed to be nonsingular. This has implications in the Cayley transform.

Example A.1. Consider the pseudo-vector = [ 6 2 3 ]T, for which = 62 + 22 + 32 = 7. Theassociated spin matrix and its square are

=

0 3 23 0 6

2 6 0

,

2 =

13 12 1812 45 6

18

6 40

. (A.16)

The eigenvalues of are (0, 7i, 7i) while those of2 are (0, 7, 7).

A.4. From Spinors To Rotators

Referring to Figure A.2, a rotator is an operator that maps a generic point P(x) to Q(x ) given the

rotation axis and the angle . We consider only rotator representations in the form ofrotationmatrices R, defined by

x = Rx. (A.17)This 3

3 matrix is proper orthogonal, that is, RTR

=I and det(R)

= +1. It must reduce to I if

the rotation vanishes. Another important attribute is the trace property

trace(R) = 1 + 2cos , (A.18)

proofs of which may be found for example in Goldstein [267, p.124] or Hammermesh [285, p.325].

The problem considered in this section is the construction ofR from the rotation data. The inverse

problem: given R, extract and , is treated in the next section. Now ifR is assumed to be analytic

in it must have the Taylor expansion R = I+ c1+ c22 + c33 + . . ., where all ci must vanishif = 0. But in view of the Cayley-Hamilton theorem (A.15) , all powers of order 3 or higher maybe eliminated, and so R must be a linear function ofI, and 2. For convenience this will be

writtenR = I + () + ()2, (A.19)

in which is the scaling factor discussed above, whereas and are scalar functions of and of the

invariants of or . Since the only invariant of the latter is , we can anticipate that = (,)and = (,), both vanishing if = 0. Two techniques to determine those coefficients for = 1 are discussed next. Table A.1 summarizes the most important representations of the rotatorin terms of the scaled .

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Table A.1. Rotator Forms for Various Spinors

Parametrization Spinor Rotator R

None 1 sin 2sin2 1

2

2 I + sin +

2sin2 12

2

2

Unit axial-vector 1 sin 2sin2 1

2 N = I + sin N + 2sin2 12 N2,Rodrigues-Cayley

tan 12

2cos

2 1

2 2cos2 1

2 B = I + 2cos2 1

2 (B + B2) = (I + B)(I B)1

DeVeubekesin 1

2

2cos1

2 2 P = I + 2cos 1

2 P + 2P2

Exponential map sin

2sin2 12

2 = I + sin

+ 2sin

2 1

2

2

2 = e = eN

A.4.1. The Algebraic Approach

This approach finds and for

=1 (the unscaled spinor) directly from algebraic conditions.

Taking the trace ofR given by (A.19) and applying the trace property (A.18) requires

3 22 = 1 + 2cos , = 1 cos 2

= 2 sin2 1

2

2. (A.20)

The orthogonality condition I = RTR = (I + 2)(I + + 2) = I + (2 2)2 +24 = I + (2 2 22)2 leads to

2 2 22 = 0 = sin

. (A.21)

Hence

R = I + sin + 1 cos

2

2 = I + sin + 2 sin

2 12

2

2. (A.22)

From a computational viewpoint the sine-squaredform should be preferred for small angles to avoid

the 1 cos cancellation. Replacing the components of and2 gives the explicit rotator form

R = 12

21 + (22 + 23) cos 212 sin2 12 3 sin 213 sin2 12 + 2 sin

212 sin2 1

2 + 3 sin 22 + (23 + 21) cos 223 sin2 12 1 sin

213 sin2 1

2 2 sin 223 sin2 12 + 1 sin 23 + (21 + 22) cos

.

(A.23)

This is invariant to scaling of the i , and consequently (A.23) is unique.

A.4.2. The Geometric Approach

The vector representation of the rigid motion pictured in Figure A.2 is

x = x cos + (n x) sin + n(n x)(1 cos ) = x + (n x) sin +n (n x) (1 cos ),

(A.24)

where n is normalized to unit length as per (A.6). This can be recast in matrix form by substitutingn x Nx = x/ and x = Rx. Cancelling x we get back (A.22).

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Identifying we get the conditions = and + 2 = . The first one gives = / ,which inserted in the second requires 2 2 +2 = 0. Fortunately this is identically satisfied by(A.21). Thus the only solution is = / = (1 cos )/( sin ) = tan 1

2/. This is precisely

the Rodrigues normalization (A.7) or (A.27). Consequently

R = (I + B)(I B)1, B =tan 1

2

. (A.31)

The explicit calculation ofR in terms of the bi leads to (A.26).

A.4.4. Exponential Map

This is a final representation ofR that has both theoretical and practical importance. Given a

skew-symmetric real matrix S, the matrix exponential

Q = eS = Exp(S) (A.32)is proper orthogonal. Here is the simple proof of Gantmacher [254, p 287]: QT = Exp(ST) =Exp(ST) = Q1. If the eigenvalues ofS are i , then i i = trace(S) = 0. The eigenvaluesofQ are i = exp(i ); thus det(Q) = i i = exp(i i ) = exp(0) = +1. The transformation(A.32) is called an exponential map. The converse is of course S = Log(Q).As in the case of the Cayley transform, one may pose the question of whether we can get the rotator

R = Exp() exactly for some factor = ( , ). To study this question we need an explicitform of the exponential. This can be obtained from (A.13) in which the function Exp so that the

diagonal matrix entries are1 and exp( i) = cos i sin . The following approach is moreinstructive and leads directly to the final result. Start from the definition of the matrix exponential

Exp() = I + + 2

2!

2 + 3

3!

3 + . . . (A.33)and use the Cayley-Hamilton theorem (A.15) to eliminate all powers of order 3 or higher in .

Identify the coefficient series of and 2 with those of the sine and cosine, to obtain

Exp() = I + sin()

+ 1 cos()2

2. (A.34)

Comparing to (A.22) requires = , or = /. Introducing i = i / and = Spin() =N = (/) as in (A.9), one gets

R = Exp() = I + sin + 1 cos

2

2 = I + sin + 2sin

2 12

2

2. (A.35)

On substituting = N this recovers R = I + sin N + (1 cos ) N2, as it should.This representation has several advantages: (i) it is singularity free, and (ii) the i are exactly

proportional to the angle, and (ii) it simplifies differentiation. Because of these favorable propertiesthe exponential map has become a favorite of implementations where large angles may occur, as in

orbiting structures and robotics.

Example A.2.

Take again = [ 3 2 6 ]T so = 7. Consider the 3 angles = 2, = 90 and = 180. The rotatorscalculated from (A.22) are, to 8 figures:

R = (A.36)

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A11 A.6 ROTATOR DERIVATIVES

A.5. From Rotators to Spinors

IfR is given, the extraction of the rotation amount and the unit pseudo-vector n = / is oftenrequired. The former is easy using the trace property (A.19):

cos

=12

(trace(R)

1) . (A.37)

Recovery ofn is also straightforward using the particular form R = I + sin N + (1 cos )N2since R RT = 2sin N, whence

N = R RT

2sin , n = axial(N). (A.38)

One issue is the sign of since (A.37) is satisfied by . If the sign is reversed, so is n. Thus inprinciple it is possible to select 0 if no constraints are placed on the direction of the rotationaxis.

The above formulas are prone to numerical instability for near 0 and 180 since sin vanishes.

A robust algorithm is that given by Spurrier [579] in the language of quaternions. Choose thealgebraically largest oftrace(R) and Ri i , i = 1, 2, 3. Iftrace(R) is the largest, computep0 = cos 12 = 12

1 + trace(R), pi = ni sin 12 = 14 (Rk j Rjk)/p0, i = 1, 2, 3, (A.39)

in which j and k are the cyclic permutations ofi . Otherwise let Ri i be the algebraically largest

diagonal entry, and again denote i, j, kthe cyclic permutation. Then use

pi = ni sin 12 =

12

Ri i + 14 (1 trace(R)), p0 = cos 12 = 14 (Rk j Rjk)/pi ,pi = 14 (Rl,i + Ri,l )/pi , l = j, k.

(A.40)

From p0

, p1

, p2

, p3

it is easy to pass to , n1, n2, n3 once the sign of is chosen as discussed

above.

Remark A.1. The theoretical formula for the matrix logarithm, which is applicable to any matrix size, is

= log R = arcsin 2

axial(R RT), (A.41)

in which = 12|axial(R RT)|. But for numerical computations this expression is largely useless.

A.6. Rotator Derivatives

The derivatives and differentials of the rotator with respect to angle and rotation axis direction

changes are required in many developments.As independent parameters we will take and = n (the ni are not good choices because theyare linked by a constraint). For convenience, define

d =

d1d2d3

, d = Spin(d), d2 =

d21d22d23

, d2 = Spin(d2),

W = cos I + sin N = R (1 cos ) nnT.(A.42)

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It is convenient to depart from R = Exp(N) = I+sin N+(1cos ) N2 = W+(1cos )nnT.Succesive differentiation gives

dR = R d(N) = R (N d + dN), (A.43)

and

d2R = R d(N) d(N) + R d2(N) = R (N d + dN)2 + N d2 + 2 d dN + d2N (A.44)

These expressions can be simplified using the following identities:

RN = NR = N + sin N2 + (1 cos )N3 = cos N + sin N2 = WN = NW,NRN = N2 R = R N2 = sin N + cos N2 = R I N2 = R nnT,

dN = d

N T

d2

N dN = dT

(T

d) I2

, N2 dN = N dT

(T

d) N2

,

R dN = Rd

WNTd

2.

(A.45)

Here Nn = 0 has been used to eliminate some terms.Inserting into (A.43) gives

dR = W N d + R dN (A.46)

This is a compacted form verified through Mathematica.The expression ofd2R, which is required for stiffness level and acceleration computations, will be

worked out later.

A.7. Spinor and Rotator Transformations

Suppose isa spinorand R = Rot() theassociatedrotator, referred toa Cartesianframe x = {xi }.It is required to transform R to another Cartesian frame x = { xi } related by Ti j = xi /xj , whereTi j are entries of a 3 3 orthogonal matrix T = x/x. Application of (?) yields

R = TRTT, RT = TRTTT, R = TTRT, R = TTRTT. (A.47)

More details may be found in [267, Chapter 4]. Pre and post-multiplying (?) by T and TT,

respectively, yields the transformed spinor N = T N TT, which is also skew-symmetric becauseN

T = T NT TT = N. Likewise for the other spinors listed in Table A.5. Relations (?) withQ T show how axial vectors transform.

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A13 A.9 SPINOR AND ROTATOR DIFFERENTIATION

A.8. Axial Vector Jacobian

The Jacobian matrix H() = / of the rotational axial vector with respect to the spin axialvector , and its inverse H()1 = /, appear in the EICR. The latter was first derived bySimo [564] and Szwabowicz [602], and rederived by Nour-Omid and Rankin [433, p. 377]:

H()1 =

= sin

I+1 cos 2

+ sin 3

T = I+1 cos 2

+ sin 3

2. (A.48)

The last expression in (A.48), not given by the cited authors, is obtained on replacing T = 2I +2. Use of the inversion formula (?) gives

H() =

= I 12+ 2, (A.49)

in which

= 1 12 cot( 12 ) 2

= 112

+ 1720

2 + 130240

4 + 11209600

6 + . . . (A.50)

The givenin (A.50) results by simplifying thevalue = [sin (1 + cos )]/[ 2 sin ] given byprevious investigators. Care must be taken on evaluating for small angle because it approaches

0/0. If| | < 1/20, say, the series given above may be used, with error < 1016 when 4 terms areretained. If is a multiple of 2 , blows up since cot( 1

2 ) , and a modulo-2 reduction is

required.

In the formulation of the tangent stiffness matrix, the spin derivative ofH()T contracted with a

nodal moment vector m is required:

L(, m) = H()T

:m =

[H()Tm] H()

= [(Tm) I + mT 2mT] + 2 mT 12Spin(m)

H().

(A.51)

in which

= d/d

= 2 + 4cos + sin 4

4 4 sin2( 12

)= 1

360+ 1

7560 2 + 1

201600 4 + 1

5987520 6 + . . . (A.52)

This expression of was obtained by simplifying results given in [433, p. 378].

A.9. Spinor and Rotator Differentiation

Derivatives, differentials and variations of axial vectors, spinors and rotators with respect to various

choices of independent variables appear in applications offinite rotations to mechanics. In this

section we present only expressions that are useful in the CR description. They are initially derived

for dynamics and then specialized to variations. Several of the formulas are new.

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A.9.1. Angular Velocities

We assume that the rotation angle (t) = (t)n(t) is a given function of time t, which is taken as theindependent variable. The time derivative of(t) is = axial(). To express rotator differentialsin a symmetric manner we introduce an axial vector and a spinor = Spin(), related to congruentially through the rotator:

= Spin() =

0 3 23 0 1

2 1 0

= RRT, = Spin() =

0 3 2

3 0 12 1 0

= RTR.

(A.53)

From (?) it follows that the axial vectors are linked by

= RT, = R. (A.54)Corresponding relation between variations or differentials, such as = R RT or d =R dRT are incorrect as shown in below. In CR dynamics, is the vector ofinertial angular

velocities whereas is the vector ofdynamic angular velocities.Repeated temporal differentiation ofR = Exp() gives the rotator time derivatives

R = R, R = (+ 2)R, ...R = (...+ 2+ + 3)R, . . . (A.55)

R = R , R = R(+ 2), ...R = R(...+ 2+ + 3), . . . (A.56)The following four groupings appear often: R R

T, R RT, RT R and R

TR. From the identities

RRT = I and RTR = I it can be shown that they are skew-symmetric, and may be associatedto axial vectors with physical meaning. For example, taking time derivatives ofRRT = I yieldsR R

T

+R RT

=0, whence R R

T

= R RT

= (R R)T. Pre and postmultiplication ofR in (A.55)

and (A.56) by R and RT

furnishes

R RT = R RT = , RT R = RTR = . (A.57)Note that the general integral ofR = R aside from a constant, is R = Exp(), from which (t)can be extracted. On the other hand there is no integral relation defining (t); only the differential

equation R = R .A.9.2. Angular Accelerations

Postmultiplying the second of (A.55) by RT yields

RRT = + 2 = + T ( )2I =

0 3 23 0 1

2 1 0

+ 22 23 12 13

12 23 21 2313 23 21 22

,

(A.58)

in which ( )2 = 21 + 22 + 23 . When applied to a vector r, (+ )r = r + T

r ( )2r.This operator appears in the expression of particle accelerations in a moving frame. The second and

third term give rise to the Coriolis and centrifugal forces, respectively. Premultiplying the second

derivative in (A.56) by R yields RR = + , and so on.

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A15 A.9 SPINOR AND ROTATOR DIFFERENTIATION

A.9.3. Variations

Some of the foregoing expressions can be directly transformed to variational and differential forms

while others cannot. For example, varying R = Exp() gives

R

=R, RT

= RT . (A.59)

This matches R = R and RT = RT from (A.55) on replacing ( ) by . On the other hand,the counterparts of (A.56): R = R and RT = R are notR = R and RT = RT, apoint that has tripped authors unfamiliar with moving frame dynamics. Correct handling requires

the introduction of a third axial vector :

R = R , RT = R, in which = Spin() =

0 3 23 0 1

2 1 0

.

(A.60)

Axial vectors and can be linked as follows. Start from R

=R and R

=R . Time

differentiate the former: R = R + R , and vary the latter: R = R + R , equate thetwo right-hand sides, premultiply by RT, replace RTR and RTR by and , respectively, and

rearrange to obtain

= + , or = + = . (A.61)The last transformation is obtained by taking the axial vectors of both sides, and using (?). It is

seen that and match if and only if and commute.

Higher variations and differentials can be obtained through similar techniques. The general rule

is: if there is a rotator integral such as R = Exp(), time derivatives can be directly converted tovariations. If no integral exists, utmost care must be exerted. Physically quantities such as and

are related to a moving frame, which makes differential relations rheonomic.

A15

nfem.appa

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