TRANSFORMATION OF THE DERIVATIVE OF A VECTOR

Consider a vector v that is being observed simultaneously from two frames Fa and Fh that have relative rotation—say Fh rotates with angular velocity ы relative to Fa, which we may regard as fixed. From (A.4,3)

The derivatives of ya and h are of course

(A.4,13)

where vai = (d/dt)(vai), and so forth. It is important to note that v„ and yb are not simply two sets of components of the same vector, but are actually two different vec­tors.

Now because Fb rotates relative to Fa, the direction cosines are changing with time, and the derivative of (A.4,3) is

ifh = L.„v„ + L h/yn

or alternatively

V, = L ahyb + L abyh

the second terms representing the effect of the rotation.

Since L must be independent of v, the matrix Lab can readily be identified by considering the case when vb is constant (see Fig. A.5.). For then, from the funda­mental definitions of derivative and cross product, the derivative of v as seen from Fa is readily shown to be

dy

— = Ы X V dt

The corresponding result from (A.4,14) is

Va = tabv„ (A.4,17)

It follows from equating (A.4,16) and (A.4,17) that

L аЬУЬ = "aVa

or

Kb^b = &аКьУь (A.4,18)

for all yb. Whence — <UaLafe

^ab^ba

Finally if the above argument is repeated with Fb considered fixed, and Fa having an­gular velocity – to, we clearly arrive at the reciprocal result

Lba = – d>bLba (A.4,19)

From (A.4,18) and (A.4,19), recalling that to is skew-symmetric so that to’ = —to, the reader can readily derive the result

From (A.4,14), (A.4,18), and (A.4,19) we have the alternative relations

Vft = Lbaxa – 6>hxh xa = L ahvh + ыаха

with two additional permutations made possible by (A.4,20). A particular form we shall finally want for application is that which uses the components of xa transformed into Fh, viz.

L bJa = + 6>bxh (A.4,22)

TRANSFORMATION OF A MATRIX

Equation (A.4,20) is an example of the transformation of a matrix, the elements of which are dependent on the frame of reference. Generally the matrix of interest A oc­curs in an equation of the form

v = Au (A.4,23)

where the elements of the (physical) vectors u and v and of the matrix A are all de­pendent on the reference frame. We write (A.4,23) for each of the two frames Fa and Fh, that is,

v„ = A„u(,

v,, = Ahub

and transform the second to

= A,,Lfc„uu

Premultiplying by Lab we get

Va = f a/)A/)L/,„U„

By comparison with (A.4,24a) we get the general result

A a = LahAbLba

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