Mathematics of Physics and Engineering

General Accelerating Frames^63

that the matrix UT is a representation, in the basis (z, j, k), of an or-

thogonal transformation that rotates the space so that (z, j, k) moves into

Now assume that the picture on Figure 2.1.9 is changing in time as
follows:

• Frame O and its coordinate system (i, j, k) are fixed (not moving).
  
  
  • Oi(t) = 0 for alii.
  
  
  • The coordinate system (?i, jx, k{) in frame 0\ is moving (rotating)
    relative to (i, j, k).
  
  
  • The point P is fixed in frame 0\ relative to (ii, jly k).
  
  
  
  

Thus, xi,yi,z\ are constants and P is rotating in the (?, j, k) frame. Then

(2.1.33) becomes

x(t) i + y(t) j + z(t) k = xi h(t) +yi3i(t) +ziki(t),

Define the matrix U = U(t) according to (2.1.38), and assume that the

entries of the matrix U are differentiable functions of time; it is a reasonable

assumption if the rotation is without jerking. Since U(t)UT(t) = I for all t,

it follows that d/dt(UUT) = 0, the zero matrix. The product rule applies

to matrix differentiation and therefore

i/uT + ui/T = iiuT + (iiuT)T = o,

which means that fi(£) = U(t)UT(t) is antisymmetric, that is, has the form

/ 0 -w 3 (i) W2(t) \

fi(t) = w 3 (t) 0 -wi(t)

We use the entries of the matrix Q(t) to define mathematically the

instantaneous rotation vector in the fixed frame O:

u(t) = u>i (t) i + W2(t)j+ w 3 (t) k. (2.1.39)

EXERCISE 2.1.17? (a) Verify that, for every vector R = Rii + R^j+R 3 k

and each t,

Q.{t) R = w(t) x R. (2.1.40)