An infinitesimal transformation of a Vector is given by
|
(1) |
where the Matrix
is infinitesimal and I is the Identity Matrix. (Note that the infinitesimal
transformation may not correspond to an inversion, since inversion is a discontinuous process.) The
Commutativity of infinitesimal transformations
and
is established by the
equivalence of
|
(2) |
|
(3) |
Now let
|
(4) |
The inverse
is then
, since
|
(5) |
Since we are defining our infinitesimal transformation to be a rotation, Orthogonality
of Rotation Matrices
requires that
|
(6) |
but
|
(7) |
|
(8) |
so
and the infinitesimal rotation is Antisymmetric. It must therefore have
a Matrix of the form
|
(9) |
The differential change in a vector upon application of the Rotation Matrix is then
|
(10) |
Writing in Matrix form,
Therefore,
|
(13) |
where
|
(14) |
The total rotation observed in the stationary frame will be a sum of the rotational velocity and the velocity in the
rotating frame. However, note that an observer in the stationary frame will see a velocity opposite in direction to that
of the observer in the frame of the rotating body, so
|
(15) |
This can be written as an operator equation, known as the Rotation Operator, defined as
|
(16) |
See also Acceleration, Euler Angles, Rotation, Rotation Matrix, Rotation Operator
© 1996-9 Eric W. Weisstein
1999-05-26