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Unformatted text preview: e the principal moments of inertia. Hence, in the body frame, the components of Equation (477) yield (478)
and (480) where . Here, we have made use of the fact that the moments of inertia of a rigid body are constant in time in the co-rotating body frame. The preceding three equations are known as Euler's equations.
Consider a body that is freely rotating--that is, in the absence of external torques. Furthermore, let the body
be rotationally symmetric about the -axis. It follows that
. Likewise, we can write
. In general, however, . Thus, Euler's equations yield (481)
and Clearly, (483) is a constant of the motion. Equation (481) and (482) can be written (484)
and http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node70.html#e9.73a (485) Page 2 of 4 Euler's equations 11/6/13 1:42 AM where . As is easily demonstrated, the solution to these equations is (486)
and where (487) is a constant. Thus, the projection of the angular velocity vector onto the fixed length , and rotates steadily about the -axis with angular velocity of the angular velocity vector,
velocity vector subtends some constant angle, - plane has the . It follows that the length , is a constant of the motion. Clearly, the angular
, with the -axis, which implies that and . Hence, the components of the angular velocity vector are (48...
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