The fixed frame and the body frame share the same origin. Hence, we can transform from one to the other by means of an appropriate rotation of our coordinate axes. In general, if we restrict ourselves to rotations about one of the Cartesian axes, three successive rotations are required to transform the fixed frame into the body frame. There are, in fact, many different ways to combined three successive rotations in order to achieve this goal. In the following, we shall describe the most widely used method, which is due to Euler.
We start in the fixed frame, which has coordinates ,
,
, and
unit vectors
,
,
. Our first rotation
is counterclockwise (if we look down the axis) through an angle
about the
-axis. The new frame has coordinates
,
,
, and
unit vectors
,
,
. According
to Section A.6, the transformation of coordinates can be represented as
follows:
The second rotation is counterclockwise (if we look down the axis) through
an angle about the
-axis. The new frame has coordinates
,
,
, and unit vectors
,
,
. By analogy with Equation (8.45), the transformation
of coordinates can be represented as follows:
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(8.47) |
The third rotation is counterclockwise (if we look down the axis) through
an angle about the
-axis. The new frame is the body frame, which has coordinates
,
,
, and unit vectors
,
,
. The transformation of coordinates can be represented as
follows:
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(8.49) |
The full transformation between the fixed frame and the body frame is rather complicated. However, the following results can easily be verified:
It follows from Equation (8.51) that
The angles ,
, and
are termed Euler angles. Each has a clear physical interpretation:
is the angle of precession
about the
-axis in the fixed frame,
is minus the angle of precession about the
-axis in the body frame, and
is the angle of inclination
between the
- and
- axes. Moreover, we can
express the components of the angular velocity vector
in the body frame entirely in terms of the Eulerian angles, and their time derivatives. [See Equations (8.54)–(8.56).]
Consider a freely rotating body that is rotationally symmetric about one axis (the -axis). In the absence of an external torque, the
angular momentum vector
is a constant of the motion. [See Equation (8.3).] Let
point along the
-axis. In the
previous section, we saw that the angular momentum vector subtends a
constant angle
with the axis of symmetry; that is, with the
-axis. Hence, the time derivative
of the Eulerian angle
is zero. We also saw that the angular momentum
vector, the axis of symmetry, and the angular velocity vector are coplanar.
Consider an instant in time at which all of these vectors lie in the
-
plane. This implies that
. According to the
previous section, the angular velocity vector subtends a constant
angle
with the symmetry axis. It follows that
and
. Equation (8.54) gives
. Hence, Equation (8.55) yields
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(8.59) |
![]() |
(8.60) |
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