(o03dn <y + /i2(to01to02/ fto3m)sn upt cn upt l-(M«W«>3»)2snV

where

From Eqs. (16-72c), (16-38), and (16-39), we also have

Equations (16-74a, b, c), are significantly more complex than Eqs. (16-60a, b, c), the analogous equations for axial symmetry. In these equations, the dependence of the Jacobian elliptic functions on the parameter has been omitted for notational convenience.

16.2.2 Torque-Free Motion—Kinematic Equations

The various forms of the kinematic equations of motion considered in Section 16.1.2 contain components of the instantaneous angular velocity vector, «, on the right-hand side. The solutions for <o obtained above can be substituted into the kinematic equations in the torque-free case; however, this leads to rather intractable differential equations, which can be avoided by a suitable choice of coordinate system. An especially convenient inertial reference system is one in which the angular momentum vector, which is fixed in inertial space if N=0, lies along the third coordinate axis. Then L in body coordinates is given by

where A is the direction cosine matrix. The most convenient kinematic parameters in this case are the 3-1-3 Euler angles, so we use Eq. (12-20) for A to obtain

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